id stringlengths 16 152 | package stringclasses 563
values | symbol stringlengths 1 125 | headline stringlengths 0 249 | usage stringlengths 0 722 | description stringclasses 11
values | example_code stringlengths 0 2.87k | example_output stringclasses 42
values | see_also stringlengths 0 1.59k | source_file stringclasses 602
values | branch stringclasses 1
value | text_for_embedding stringlengths 0 4.44k | raw_text stringlengths 14 77.1k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
doc_534a7610_(rightMultiplicationMap,RingElement,ZZ,ZZ) | doc | (rightMultiplicationMap,RingElement,ZZ,ZZ) | Computes a matrix for left or right multiplication by a homogeneous element | leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis) | C = QQ<|x,y|>
D = C/ideal{x^2+x*y,y^2}
isRightRegular(x,1)
L = leftMultiplicationMap(x,1)
M=matrix gens kernel L
ncBasis(1,D)*M | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes a matrix for left or right multiplication by a homogeneous element
USAGE: leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis)
INPUTS: r : RingElement
n : ZZ
the homogeneous degree for the source of the map
m : ZZ
th... | Key
leftMultiplicationMap
(leftMultiplicationMap,RingElement,ZZ)
(leftMultiplicationMap,RingElement,ZZ,ZZ)
(leftMultiplicationMap,RingElement,List,List)
rightMultiplicationMap
(rightMultiplicationMap,RingElement,ZZ)
(rightMultiplicationMap,RingElement,ZZ,ZZ)
(rightMultipl... | |||
doc_534a7610_(rightMultiplicationMap,RingElement,List,List) | doc | (rightMultiplicationMap,RingElement,List,List) | Computes a matrix for left or right multiplication by a homogeneous element | leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis) | C = QQ<|x,y|>
D = C/ideal{x^2+x*y,y^2}
isRightRegular(x,1)
L = leftMultiplicationMap(x,1)
M=matrix gens kernel L
ncBasis(1,D)*M | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes a matrix for left or right multiplication by a homogeneous element
USAGE: leftMultiplicationMap(r,n) or leftMultiplicationMap(r,n,m) or leftMultiplicationMap(r,fromBasis,toBasis)
INPUTS: r : RingElement
n : ZZ
the homogeneous degree for the source of the map
m : ZZ
th... | Key
leftMultiplicationMap
(leftMultiplicationMap,RingElement,ZZ)
(leftMultiplicationMap,RingElement,ZZ,ZZ)
(leftMultiplicationMap,RingElement,List,List)
rightMultiplicationMap
(rightMultiplicationMap,RingElement,ZZ)
(rightMultiplicationMap,RingElement,ZZ,ZZ)
(rightMultipl... | |||
doc_e4261e09_isLeftRegular | doc | isLeftRegular | Determines if a given (homogeneous) element is regular in a given degree | isLeftRegular(x,n) or isRightRegular(x,n) | B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
g = z^3 + y*z*x - z*y*x - y^3
isLeftRegular(g,6)
C = QQ<|x,y|>
D = C/ideal{x^2+x*y,y^2}
isLeftRegular(x,1)
isRightRegular(x,1) | leftMultiplicationMap
rightMultiplicationMap | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Determines if a given (homogeneous) element is regular in a given degree
USAGE: isLeftRegular(x,n) or isRightRegular(x,n)
INPUTS: x : RingElement
n : ZZ
the degree in which regularity is checked.
OUTPUTS: : Boolean
EXAMPLE CODE:
```macaulay2
B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
g = ... | Key
isLeftRegular
(isLeftRegular,RingElement,ZZ)
isRightRegular
(isRightRegular,RingElement,ZZ)
Headline
Determines if a given (homogeneous) element is regular in a given degree
Usage
isLeftRegular(x,n) or isRightRegular(x,n)
Inputs
x : RingElement
n : ZZ
... | ||
doc_e4261e09_(isLeftRegular,RingElement,ZZ) | doc | (isLeftRegular,RingElement,ZZ) | Determines if a given (homogeneous) element is regular in a given degree | isLeftRegular(x,n) or isRightRegular(x,n) | B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
g = z^3 + y*z*x - z*y*x - y^3
isLeftRegular(g,6)
C = QQ<|x,y|>
D = C/ideal{x^2+x*y,y^2}
isLeftRegular(x,1)
isRightRegular(x,1) | leftMultiplicationMap
rightMultiplicationMap | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Determines if a given (homogeneous) element is regular in a given degree
USAGE: isLeftRegular(x,n) or isRightRegular(x,n)
INPUTS: x : RingElement
n : ZZ
the degree in which regularity is checked.
OUTPUTS: : Boolean
EXAMPLE CODE:
```macaulay2
B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
g = ... | Key
isLeftRegular
(isLeftRegular,RingElement,ZZ)
isRightRegular
(isRightRegular,RingElement,ZZ)
Headline
Determines if a given (homogeneous) element is regular in a given degree
Usage
isLeftRegular(x,n) or isRightRegular(x,n)
Inputs
x : RingElement
n : ZZ
... | ||
doc_e4261e09_isRightRegular | doc | isRightRegular | Determines if a given (homogeneous) element is regular in a given degree | isLeftRegular(x,n) or isRightRegular(x,n) | B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
g = z^3 + y*z*x - z*y*x - y^3
isLeftRegular(g,6)
C = QQ<|x,y|>
D = C/ideal{x^2+x*y,y^2}
isLeftRegular(x,1)
isRightRegular(x,1) | leftMultiplicationMap
rightMultiplicationMap | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Determines if a given (homogeneous) element is regular in a given degree
USAGE: isLeftRegular(x,n) or isRightRegular(x,n)
INPUTS: x : RingElement
n : ZZ
the degree in which regularity is checked.
OUTPUTS: : Boolean
EXAMPLE CODE:
```macaulay2
B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
g = ... | Key
isLeftRegular
(isLeftRegular,RingElement,ZZ)
isRightRegular
(isRightRegular,RingElement,ZZ)
Headline
Determines if a given (homogeneous) element is regular in a given degree
Usage
isLeftRegular(x,n) or isRightRegular(x,n)
Inputs
x : RingElement
n : ZZ
... | ||
doc_e4261e09_(isRightRegular,RingElement,ZZ) | doc | (isRightRegular,RingElement,ZZ) | Determines if a given (homogeneous) element is regular in a given degree | isLeftRegular(x,n) or isRightRegular(x,n) | B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
g = z^3 + y*z*x - z*y*x - y^3
isLeftRegular(g,6)
C = QQ<|x,y|>
D = C/ideal{x^2+x*y,y^2}
isLeftRegular(x,1)
isRightRegular(x,1) | leftMultiplicationMap
rightMultiplicationMap | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Determines if a given (homogeneous) element is regular in a given degree
USAGE: isLeftRegular(x,n) or isRightRegular(x,n)
INPUTS: x : RingElement
n : ZZ
the degree in which regularity is checked.
OUTPUTS: : Boolean
EXAMPLE CODE:
```macaulay2
B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
g = ... | Key
isLeftRegular
(isLeftRegular,RingElement,ZZ)
isRightRegular
(isRightRegular,RingElement,ZZ)
Headline
Determines if a given (homogeneous) element is regular in a given degree
Usage
isLeftRegular(x,n) or isRightRegular(x,n)
Inputs
x : RingElement
n : ZZ
... | ||
doc_cdf48f11_isCentral | doc | isCentral | Determines if an element is central | isCentral x or isCentral(x,ncgb) | B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
g = z^3 + y*z*x - z*y*x - y^3
h = x^2 + y^2 + z^2
isCentral h
isCentral g | centralElements | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Determines if an element is central
USAGE: isCentral x or isCentral(x,ncgb)
INPUTS: x : RingElement
OUTPUTS: : Boolean
EXAMPLE CODE:
```macaulay2
B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
g = z^3 + y*z*x - z*y*x - y^3
h = x^2 + y^2 + z^2
isCentral h
isCentral g
```
SEEALSO: centralEleme... | Key
isCentral
(isCentral,RingElement)
Headline
Determines if an element is central
Usage
isCentral x or isCentral(x,ncgb)
Inputs
x : RingElement
Outputs
: Boolean
Description
Text
This method checks to see if a given noncommutative ring element is centra... | ||
doc_cdf48f11_(isCentral,RingElement) | doc | (isCentral,RingElement) | Determines if an element is central | isCentral x or isCentral(x,ncgb) | B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
g = z^3 + y*z*x - z*y*x - y^3
h = x^2 + y^2 + z^2
isCentral h
isCentral g | centralElements | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Determines if an element is central
USAGE: isCentral x or isCentral(x,ncgb)
INPUTS: x : RingElement
OUTPUTS: : Boolean
EXAMPLE CODE:
```macaulay2
B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
g = z^3 + y*z*x - z*y*x - y^3
h = x^2 + y^2 + z^2
isCentral h
isCentral g
```
SEEALSO: centralEleme... | Key
isCentral
(isCentral,RingElement)
Headline
Determines if an element is central
Usage
isCentral x or isCentral(x,ncgb)
Inputs
x : RingElement
Outputs
: Boolean
Description
Text
This method checks to see if a given noncommutative ring element is centra... | ||
doc_66e7cd5e_centralElements | doc | centralElements | Finds central elements in a given degree | centralElements(A,n) | B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
centralElements(B,2)
centralElements(B,3) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Finds central elements in a given degree
USAGE: centralElements(A,n)
INPUTS: A : Ring
n : ZZ
the homogeneous degree in which to compute central elements
OUTPUTS: : Matrix
EXAMPLE CODE:
```macaulay2
B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
centralElements(B,2)
centralElements(B,3)... | Key
centralElements
(centralElements, Ring, ZZ)
Headline
Finds central elements in a given degree
Usage
centralElements(A,n)
Inputs
A : Ring
n : ZZ
the homogeneous degree in which to compute central elements
Outputs
: Matrix
Description
Text
... | |||
doc_66e7cd5e_(centralElements,_Ring,_ZZ) | doc | (centralElements, Ring, ZZ) | Finds central elements in a given degree | centralElements(A,n) | B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
centralElements(B,2)
centralElements(B,3) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Finds central elements in a given degree
USAGE: centralElements(A,n)
INPUTS: A : Ring
n : ZZ
the homogeneous degree in which to compute central elements
OUTPUTS: : Matrix
EXAMPLE CODE:
```macaulay2
B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
centralElements(B,2)
centralElements(B,3)... | Key
centralElements
(centralElements, Ring, ZZ)
Headline
Finds central elements in a given degree
Usage
centralElements(A,n)
Inputs
A : Ring
n : ZZ
the homogeneous degree in which to compute central elements
Outputs
: Matrix
Description
Text
... | |||
doc_24a5eb61_oreExtension | doc | oreExtension | Creates an Ore extension of a noncommutative ring | oreExtension(A,sigma,delta,x) or oreExtension(A,sigma,x) | B = toFreeAlgebraQuotient(QQ[x])
sigma = map(B,B,{x})
delta = derivation(B,{1_B})
C = oreExtension(B,sigma,delta,dx) | oreIdeal | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates an Ore extension of a noncommutative ring
USAGE: oreExtension(A,sigma,delta,x) or oreExtension(A,sigma,x)
INPUTS: A : Ring
sigma : RingMap
delta : Derivation
x : RingElement
or a @ TO Symbol @
OUTPUTS: : QuotientRing
EXAMPLE CODE:
```macaulay2
B = toFreeAlgebraQuotient(... | Key
oreExtension
(oreExtension,Ring,RingMap,Derivation,RingElement)
(oreExtension,Ring,RingMap,Derivation,Symbol)
(oreExtension,Ring,RingMap,RingElement)
(oreExtension,Ring,RingMap,Symbol)
[oreExtension, Degree]
Headline
Creates an Ore extension of a noncommutative ring
U... | ||
doc_24a5eb61_(oreExtension,Ring,RingMap,Derivation,RingElement) | doc | (oreExtension,Ring,RingMap,Derivation,RingElement) | Creates an Ore extension of a noncommutative ring | oreExtension(A,sigma,delta,x) or oreExtension(A,sigma,x) | B = toFreeAlgebraQuotient(QQ[x])
sigma = map(B,B,{x})
delta = derivation(B,{1_B})
C = oreExtension(B,sigma,delta,dx) | oreIdeal | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates an Ore extension of a noncommutative ring
USAGE: oreExtension(A,sigma,delta,x) or oreExtension(A,sigma,x)
INPUTS: A : Ring
sigma : RingMap
delta : Derivation
x : RingElement
or a @ TO Symbol @
OUTPUTS: : QuotientRing
EXAMPLE CODE:
```macaulay2
B = toFreeAlgebraQuotient(... | Key
oreExtension
(oreExtension,Ring,RingMap,Derivation,RingElement)
(oreExtension,Ring,RingMap,Derivation,Symbol)
(oreExtension,Ring,RingMap,RingElement)
(oreExtension,Ring,RingMap,Symbol)
[oreExtension, Degree]
Headline
Creates an Ore extension of a noncommutative ring
U... | ||
doc_24a5eb61_(oreExtension,Ring,RingMap,Derivation,Symbol) | doc | (oreExtension,Ring,RingMap,Derivation,Symbol) | Creates an Ore extension of a noncommutative ring | oreExtension(A,sigma,delta,x) or oreExtension(A,sigma,x) | B = toFreeAlgebraQuotient(QQ[x])
sigma = map(B,B,{x})
delta = derivation(B,{1_B})
C = oreExtension(B,sigma,delta,dx) | oreIdeal | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates an Ore extension of a noncommutative ring
USAGE: oreExtension(A,sigma,delta,x) or oreExtension(A,sigma,x)
INPUTS: A : Ring
sigma : RingMap
delta : Derivation
x : RingElement
or a @ TO Symbol @
OUTPUTS: : QuotientRing
EXAMPLE CODE:
```macaulay2
B = toFreeAlgebraQuotient(... | Key
oreExtension
(oreExtension,Ring,RingMap,Derivation,RingElement)
(oreExtension,Ring,RingMap,Derivation,Symbol)
(oreExtension,Ring,RingMap,RingElement)
(oreExtension,Ring,RingMap,Symbol)
[oreExtension, Degree]
Headline
Creates an Ore extension of a noncommutative ring
U... | ||
doc_24a5eb61_(oreExtension,Ring,RingMap,RingElement) | doc | (oreExtension,Ring,RingMap,RingElement) | Creates an Ore extension of a noncommutative ring | oreExtension(A,sigma,delta,x) or oreExtension(A,sigma,x) | B = toFreeAlgebraQuotient(QQ[x])
sigma = map(B,B,{x})
delta = derivation(B,{1_B})
C = oreExtension(B,sigma,delta,dx) | oreIdeal | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates an Ore extension of a noncommutative ring
USAGE: oreExtension(A,sigma,delta,x) or oreExtension(A,sigma,x)
INPUTS: A : Ring
sigma : RingMap
delta : Derivation
x : RingElement
or a @ TO Symbol @
OUTPUTS: : QuotientRing
EXAMPLE CODE:
```macaulay2
B = toFreeAlgebraQuotient(... | Key
oreExtension
(oreExtension,Ring,RingMap,Derivation,RingElement)
(oreExtension,Ring,RingMap,Derivation,Symbol)
(oreExtension,Ring,RingMap,RingElement)
(oreExtension,Ring,RingMap,Symbol)
[oreExtension, Degree]
Headline
Creates an Ore extension of a noncommutative ring
U... | ||
doc_24a5eb61_(oreExtension,Ring,RingMap,Symbol) | doc | (oreExtension,Ring,RingMap,Symbol) | Creates an Ore extension of a noncommutative ring | oreExtension(A,sigma,delta,x) or oreExtension(A,sigma,x) | B = toFreeAlgebraQuotient(QQ[x])
sigma = map(B,B,{x})
delta = derivation(B,{1_B})
C = oreExtension(B,sigma,delta,dx) | oreIdeal | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates an Ore extension of a noncommutative ring
USAGE: oreExtension(A,sigma,delta,x) or oreExtension(A,sigma,x)
INPUTS: A : Ring
sigma : RingMap
delta : Derivation
x : RingElement
or a @ TO Symbol @
OUTPUTS: : QuotientRing
EXAMPLE CODE:
```macaulay2
B = toFreeAlgebraQuotient(... | Key
oreExtension
(oreExtension,Ring,RingMap,Derivation,RingElement)
(oreExtension,Ring,RingMap,Derivation,Symbol)
(oreExtension,Ring,RingMap,RingElement)
(oreExtension,Ring,RingMap,Symbol)
[oreExtension, Degree]
Headline
Creates an Ore extension of a noncommutative ring
U... | ||
doc_24a5eb61_[oreExtension,_Degree] | doc | [oreExtension, Degree] | Creates an Ore extension of a noncommutative ring | oreExtension(A,sigma,delta,x) or oreExtension(A,sigma,x) | B = toFreeAlgebraQuotient(QQ[x])
sigma = map(B,B,{x})
delta = derivation(B,{1_B})
C = oreExtension(B,sigma,delta,dx) | oreIdeal | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates an Ore extension of a noncommutative ring
USAGE: oreExtension(A,sigma,delta,x) or oreExtension(A,sigma,x)
INPUTS: A : Ring
sigma : RingMap
delta : Derivation
x : RingElement
or a @ TO Symbol @
OUTPUTS: : QuotientRing
EXAMPLE CODE:
```macaulay2
B = toFreeAlgebraQuotient(... | Key
oreExtension
(oreExtension,Ring,RingMap,Derivation,RingElement)
(oreExtension,Ring,RingMap,Derivation,Symbol)
(oreExtension,Ring,RingMap,RingElement)
(oreExtension,Ring,RingMap,Symbol)
[oreExtension, Degree]
Headline
Creates an Ore extension of a noncommutative ring
U... | ||
doc_046cc335_oreIdeal | doc | oreIdeal | Creates the defining ideal of an Ore extension of a noncommutative ring | oreIdeal(A,sigma,delta,x) or oreIdeal(A,sigma,x) | B = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w})
sigma = map(B,B,{y,z,w,x})
C = oreIdeal(B,sigma,a) | oreExtension | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates the defining ideal of an Ore extension of a noncommutative ring
USAGE: oreIdeal(A,sigma,delta,x) or oreIdeal(A,sigma,x)
INPUTS: A : Ring
sigma : RingMap
delta : Derivation
x : RingElement
or a @ TO Symbol @
OUTPUTS: : Ideal
EXAMPLE CODE:
```macaulay2
B = skewPolynomialR... | Key
oreIdeal
(oreIdeal,Ring,RingMap,Derivation,RingElement)
(oreIdeal,Ring,RingMap,Derivation,Symbol)
(oreIdeal,Ring,RingMap,RingElement)
(oreIdeal,Ring,RingMap,Symbol)
[oreIdeal, Degree]
Headline
Creates the defining ideal of an Ore extension of a noncommutative ring
Usa... | ||
doc_046cc335_(oreIdeal,Ring,RingMap,Derivation,RingElement) | doc | (oreIdeal,Ring,RingMap,Derivation,RingElement) | Creates the defining ideal of an Ore extension of a noncommutative ring | oreIdeal(A,sigma,delta,x) or oreIdeal(A,sigma,x) | B = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w})
sigma = map(B,B,{y,z,w,x})
C = oreIdeal(B,sigma,a) | oreExtension | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates the defining ideal of an Ore extension of a noncommutative ring
USAGE: oreIdeal(A,sigma,delta,x) or oreIdeal(A,sigma,x)
INPUTS: A : Ring
sigma : RingMap
delta : Derivation
x : RingElement
or a @ TO Symbol @
OUTPUTS: : Ideal
EXAMPLE CODE:
```macaulay2
B = skewPolynomialR... | Key
oreIdeal
(oreIdeal,Ring,RingMap,Derivation,RingElement)
(oreIdeal,Ring,RingMap,Derivation,Symbol)
(oreIdeal,Ring,RingMap,RingElement)
(oreIdeal,Ring,RingMap,Symbol)
[oreIdeal, Degree]
Headline
Creates the defining ideal of an Ore extension of a noncommutative ring
Usa... | ||
doc_046cc335_(oreIdeal,Ring,RingMap,Derivation,Symbol) | doc | (oreIdeal,Ring,RingMap,Derivation,Symbol) | Creates the defining ideal of an Ore extension of a noncommutative ring | oreIdeal(A,sigma,delta,x) or oreIdeal(A,sigma,x) | B = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w})
sigma = map(B,B,{y,z,w,x})
C = oreIdeal(B,sigma,a) | oreExtension | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates the defining ideal of an Ore extension of a noncommutative ring
USAGE: oreIdeal(A,sigma,delta,x) or oreIdeal(A,sigma,x)
INPUTS: A : Ring
sigma : RingMap
delta : Derivation
x : RingElement
or a @ TO Symbol @
OUTPUTS: : Ideal
EXAMPLE CODE:
```macaulay2
B = skewPolynomialR... | Key
oreIdeal
(oreIdeal,Ring,RingMap,Derivation,RingElement)
(oreIdeal,Ring,RingMap,Derivation,Symbol)
(oreIdeal,Ring,RingMap,RingElement)
(oreIdeal,Ring,RingMap,Symbol)
[oreIdeal, Degree]
Headline
Creates the defining ideal of an Ore extension of a noncommutative ring
Usa... | ||
doc_046cc335_(oreIdeal,Ring,RingMap,RingElement) | doc | (oreIdeal,Ring,RingMap,RingElement) | Creates the defining ideal of an Ore extension of a noncommutative ring | oreIdeal(A,sigma,delta,x) or oreIdeal(A,sigma,x) | B = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w})
sigma = map(B,B,{y,z,w,x})
C = oreIdeal(B,sigma,a) | oreExtension | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates the defining ideal of an Ore extension of a noncommutative ring
USAGE: oreIdeal(A,sigma,delta,x) or oreIdeal(A,sigma,x)
INPUTS: A : Ring
sigma : RingMap
delta : Derivation
x : RingElement
or a @ TO Symbol @
OUTPUTS: : Ideal
EXAMPLE CODE:
```macaulay2
B = skewPolynomialR... | Key
oreIdeal
(oreIdeal,Ring,RingMap,Derivation,RingElement)
(oreIdeal,Ring,RingMap,Derivation,Symbol)
(oreIdeal,Ring,RingMap,RingElement)
(oreIdeal,Ring,RingMap,Symbol)
[oreIdeal, Degree]
Headline
Creates the defining ideal of an Ore extension of a noncommutative ring
Usa... | ||
doc_046cc335_(oreIdeal,Ring,RingMap,Symbol) | doc | (oreIdeal,Ring,RingMap,Symbol) | Creates the defining ideal of an Ore extension of a noncommutative ring | oreIdeal(A,sigma,delta,x) or oreIdeal(A,sigma,x) | B = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w})
sigma = map(B,B,{y,z,w,x})
C = oreIdeal(B,sigma,a) | oreExtension | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates the defining ideal of an Ore extension of a noncommutative ring
USAGE: oreIdeal(A,sigma,delta,x) or oreIdeal(A,sigma,x)
INPUTS: A : Ring
sigma : RingMap
delta : Derivation
x : RingElement
or a @ TO Symbol @
OUTPUTS: : Ideal
EXAMPLE CODE:
```macaulay2
B = skewPolynomialR... | Key
oreIdeal
(oreIdeal,Ring,RingMap,Derivation,RingElement)
(oreIdeal,Ring,RingMap,Derivation,Symbol)
(oreIdeal,Ring,RingMap,RingElement)
(oreIdeal,Ring,RingMap,Symbol)
[oreIdeal, Degree]
Headline
Creates the defining ideal of an Ore extension of a noncommutative ring
Usa... | ||
doc_046cc335_[oreIdeal,_Degree] | doc | [oreIdeal, Degree] | Creates the defining ideal of an Ore extension of a noncommutative ring | oreIdeal(A,sigma,delta,x) or oreIdeal(A,sigma,x) | B = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w})
sigma = map(B,B,{y,z,w,x})
C = oreIdeal(B,sigma,a) | oreExtension | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates the defining ideal of an Ore extension of a noncommutative ring
USAGE: oreIdeal(A,sigma,delta,x) or oreIdeal(A,sigma,x)
INPUTS: A : Ring
sigma : RingMap
delta : Derivation
x : RingElement
or a @ TO Symbol @
OUTPUTS: : Ideal
EXAMPLE CODE:
```macaulay2
B = skewPolynomialR... | Key
oreIdeal
(oreIdeal,Ring,RingMap,Derivation,RingElement)
(oreIdeal,Ring,RingMap,Derivation,Symbol)
(oreIdeal,Ring,RingMap,RingElement)
(oreIdeal,Ring,RingMap,Symbol)
[oreIdeal, Degree]
Headline
Creates the defining ideal of an Ore extension of a noncommutative ring
Usa... | ||
doc_4076e47b_threeDimSklyanin | doc | threeDimSklyanin | Defines a three-dimensional Sklyanin with given parameters | threeDimSklyanin(R,params,varList) | apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> binomial(i+2,2)) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Defines a three-dimensional Sklyanin with given parameters
USAGE: threeDimSklyanin(R,params,varList)
INPUTS: R : Ring
params : List
varList : List
DegreeLimit => ZZ
OUTPUTS: : Ring
EXAMPLE CODE:
```macaulay2
apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> binomial(i+2,2... | Key
threeDimSklyanin
(threeDimSklyanin,Ring,List)
(threeDimSklyanin,Ring,List,List)
[threeDimSklyanin,DegreeLimit]
Headline
Defines a three-dimensional Sklyanin with given parameters
Usage
threeDimSklyanin(R,params,varList)
Inputs
R : Ring
params : List
... | |||
doc_4076e47b_(threeDimSklyanin,Ring,List) | doc | (threeDimSklyanin,Ring,List) | Defines a three-dimensional Sklyanin with given parameters | threeDimSklyanin(R,params,varList) | apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> binomial(i+2,2)) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Defines a three-dimensional Sklyanin with given parameters
USAGE: threeDimSklyanin(R,params,varList)
INPUTS: R : Ring
params : List
varList : List
DegreeLimit => ZZ
OUTPUTS: : Ring
EXAMPLE CODE:
```macaulay2
apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> binomial(i+2,2... | Key
threeDimSklyanin
(threeDimSklyanin,Ring,List)
(threeDimSklyanin,Ring,List,List)
[threeDimSklyanin,DegreeLimit]
Headline
Defines a three-dimensional Sklyanin with given parameters
Usage
threeDimSklyanin(R,params,varList)
Inputs
R : Ring
params : List
... | |||
doc_4076e47b_(threeDimSklyanin,Ring,List,List) | doc | (threeDimSklyanin,Ring,List,List) | Defines a three-dimensional Sklyanin with given parameters | threeDimSklyanin(R,params,varList) | apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> binomial(i+2,2)) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Defines a three-dimensional Sklyanin with given parameters
USAGE: threeDimSklyanin(R,params,varList)
INPUTS: R : Ring
params : List
varList : List
DegreeLimit => ZZ
OUTPUTS: : Ring
EXAMPLE CODE:
```macaulay2
apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> binomial(i+2,2... | Key
threeDimSklyanin
(threeDimSklyanin,Ring,List)
(threeDimSklyanin,Ring,List,List)
[threeDimSklyanin,DegreeLimit]
Headline
Defines a three-dimensional Sklyanin with given parameters
Usage
threeDimSklyanin(R,params,varList)
Inputs
R : Ring
params : List
... | |||
doc_4076e47b_[threeDimSklyanin,DegreeLimit] | doc | [threeDimSklyanin,DegreeLimit] | Defines a three-dimensional Sklyanin with given parameters | threeDimSklyanin(R,params,varList) | apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> binomial(i+2,2)) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Defines a three-dimensional Sklyanin with given parameters
USAGE: threeDimSklyanin(R,params,varList)
INPUTS: R : Ring
params : List
varList : List
DegreeLimit => ZZ
OUTPUTS: : Ring
EXAMPLE CODE:
```macaulay2
apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> binomial(i+2,2... | Key
threeDimSklyanin
(threeDimSklyanin,Ring,List)
(threeDimSklyanin,Ring,List,List)
[threeDimSklyanin,DegreeLimit]
Headline
Defines a three-dimensional Sklyanin with given parameters
Usage
threeDimSklyanin(R,params,varList)
Inputs
R : Ring
params : List
... | |||
doc_14da52f1_fourDimSklyanin | doc | fourDimSklyanin | Defines a four-dimensional Sklyanin with given parameters | fourDimSklyanin(R,params,varList) | apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> binomial(i+3,3)) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Defines a four-dimensional Sklyanin with given parameters
USAGE: fourDimSklyanin(R,params,varList)
INPUTS: R : Ring
params : List
varList : List
DegreeLimit => ZZ
OUTPUTS: : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> b... | Key
fourDimSklyanin
(fourDimSklyanin,Ring,List)
(fourDimSklyanin,Ring,List,List)
[fourDimSklyanin,DegreeLimit]
Headline
Defines a four-dimensional Sklyanin with given parameters
Usage
fourDimSklyanin(R,params,varList)
Inputs
R : Ring
params : List
va... | |||
doc_14da52f1_(fourDimSklyanin,Ring,List) | doc | (fourDimSklyanin,Ring,List) | Defines a four-dimensional Sklyanin with given parameters | fourDimSklyanin(R,params,varList) | apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> binomial(i+3,3)) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Defines a four-dimensional Sklyanin with given parameters
USAGE: fourDimSklyanin(R,params,varList)
INPUTS: R : Ring
params : List
varList : List
DegreeLimit => ZZ
OUTPUTS: : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> b... | Key
fourDimSklyanin
(fourDimSklyanin,Ring,List)
(fourDimSklyanin,Ring,List,List)
[fourDimSklyanin,DegreeLimit]
Headline
Defines a four-dimensional Sklyanin with given parameters
Usage
fourDimSklyanin(R,params,varList)
Inputs
R : Ring
params : List
va... | |||
doc_14da52f1_(fourDimSklyanin,Ring,List,List) | doc | (fourDimSklyanin,Ring,List,List) | Defines a four-dimensional Sklyanin with given parameters | fourDimSklyanin(R,params,varList) | apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> binomial(i+3,3)) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Defines a four-dimensional Sklyanin with given parameters
USAGE: fourDimSklyanin(R,params,varList)
INPUTS: R : Ring
params : List
varList : List
DegreeLimit => ZZ
OUTPUTS: : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> b... | Key
fourDimSklyanin
(fourDimSklyanin,Ring,List)
(fourDimSklyanin,Ring,List,List)
[fourDimSklyanin,DegreeLimit]
Headline
Defines a four-dimensional Sklyanin with given parameters
Usage
fourDimSklyanin(R,params,varList)
Inputs
R : Ring
params : List
va... | |||
doc_14da52f1_[fourDimSklyanin,DegreeLimit] | doc | [fourDimSklyanin,DegreeLimit] | Defines a four-dimensional Sklyanin with given parameters | fourDimSklyanin(R,params,varList) | apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> binomial(i+3,3)) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Defines a four-dimensional Sklyanin with given parameters
USAGE: fourDimSklyanin(R,params,varList)
INPUTS: R : Ring
params : List
varList : List
DegreeLimit => ZZ
OUTPUTS: : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
apply(8, i -> numgens source ncBasis(i,C))
apply(8, i -> b... | Key
fourDimSklyanin
(fourDimSklyanin,Ring,List)
(fourDimSklyanin,Ring,List,List)
[fourDimSklyanin,DegreeLimit]
Headline
Defines a four-dimensional Sklyanin with given parameters
Usage
fourDimSklyanin(R,params,varList)
Inputs
R : Ring
params : List
va... | |||
doc_31f16372_toCommRing | doc | toCommRing | Compute the abelianization of a Ring and returns a Ring. | S = toCommRing R | A = skewPolynomialRing(QQ,(-1)_QQ,{w,x,y,z})
x*y-y*x
w^2
B = toCommRing(A)
x*y
w^2
C = toCommRing(A,SkewCommutative=>true)
x*y-y*x
w^2 | toFreeAlgebraQuotient | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the abelianization of a Ring and returns a Ring.
USAGE: S = toCommRing R
INPUTS: R : FreeAlgebraQuotient
or @ TO FreeAlgebra @
SkewCommutative => Boolean
OUTPUTS: S : Ring
EXAMPLE CODE:
```macaulay2
A = skewPolynomialRing(QQ,(-1)_QQ,{w,x,y,z})
x*y-y*x
w^2
B = toCommRin... | Key
toCommRing
(toCommRing,FreeAlgebra)
(toCommRing,FreeAlgebraQuotient)
[toCommRing,SkewCommutative]
Headline
Compute the abelianization of a Ring and returns a Ring.
Usage
S = toCommRing R
Inputs
R : FreeAlgebraQuotient
or @ TO FreeAlgebra @
SkewCommut... | ||
doc_31f16372_(toCommRing,FreeAlgebra) | doc | (toCommRing,FreeAlgebra) | Compute the abelianization of a Ring and returns a Ring. | S = toCommRing R | A = skewPolynomialRing(QQ,(-1)_QQ,{w,x,y,z})
x*y-y*x
w^2
B = toCommRing(A)
x*y
w^2
C = toCommRing(A,SkewCommutative=>true)
x*y-y*x
w^2 | toFreeAlgebraQuotient | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the abelianization of a Ring and returns a Ring.
USAGE: S = toCommRing R
INPUTS: R : FreeAlgebraQuotient
or @ TO FreeAlgebra @
SkewCommutative => Boolean
OUTPUTS: S : Ring
EXAMPLE CODE:
```macaulay2
A = skewPolynomialRing(QQ,(-1)_QQ,{w,x,y,z})
x*y-y*x
w^2
B = toCommRin... | Key
toCommRing
(toCommRing,FreeAlgebra)
(toCommRing,FreeAlgebraQuotient)
[toCommRing,SkewCommutative]
Headline
Compute the abelianization of a Ring and returns a Ring.
Usage
S = toCommRing R
Inputs
R : FreeAlgebraQuotient
or @ TO FreeAlgebra @
SkewCommut... | ||
doc_31f16372_(toCommRing,FreeAlgebraQuotient) | doc | (toCommRing,FreeAlgebraQuotient) | Compute the abelianization of a Ring and returns a Ring. | S = toCommRing R | A = skewPolynomialRing(QQ,(-1)_QQ,{w,x,y,z})
x*y-y*x
w^2
B = toCommRing(A)
x*y
w^2
C = toCommRing(A,SkewCommutative=>true)
x*y-y*x
w^2 | toFreeAlgebraQuotient | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the abelianization of a Ring and returns a Ring.
USAGE: S = toCommRing R
INPUTS: R : FreeAlgebraQuotient
or @ TO FreeAlgebra @
SkewCommutative => Boolean
OUTPUTS: S : Ring
EXAMPLE CODE:
```macaulay2
A = skewPolynomialRing(QQ,(-1)_QQ,{w,x,y,z})
x*y-y*x
w^2
B = toCommRin... | Key
toCommRing
(toCommRing,FreeAlgebra)
(toCommRing,FreeAlgebraQuotient)
[toCommRing,SkewCommutative]
Headline
Compute the abelianization of a Ring and returns a Ring.
Usage
S = toCommRing R
Inputs
R : FreeAlgebraQuotient
or @ TO FreeAlgebra @
SkewCommut... | ||
doc_31f16372_[toCommRing,SkewCommutative] | doc | [toCommRing,SkewCommutative] | Compute the abelianization of a Ring and returns a Ring. | S = toCommRing R | A = skewPolynomialRing(QQ,(-1)_QQ,{w,x,y,z})
x*y-y*x
w^2
B = toCommRing(A)
x*y
w^2
C = toCommRing(A,SkewCommutative=>true)
x*y-y*x
w^2 | toFreeAlgebraQuotient | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the abelianization of a Ring and returns a Ring.
USAGE: S = toCommRing R
INPUTS: R : FreeAlgebraQuotient
or @ TO FreeAlgebra @
SkewCommutative => Boolean
OUTPUTS: S : Ring
EXAMPLE CODE:
```macaulay2
A = skewPolynomialRing(QQ,(-1)_QQ,{w,x,y,z})
x*y-y*x
w^2
B = toCommRin... | Key
toCommRing
(toCommRing,FreeAlgebra)
(toCommRing,FreeAlgebraQuotient)
[toCommRing,SkewCommutative]
Headline
Compute the abelianization of a Ring and returns a Ring.
Usage
S = toCommRing R
Inputs
R : FreeAlgebraQuotient
or @ TO FreeAlgebra @
SkewCommut... | ||
doc_c5489f44_toFreeAlgebraQuotient | doc | toFreeAlgebraQuotient | Converts a Ring to a noncommutative ring | S = toFreeAlgebraQuotient R | R = QQ[a,b,c,d, SkewCommutative=>{2,3}]
I = ideal(a*d-b*c)
S = R/I
S' = toFreeAlgebraQuotient(S)
ideal S' | toCommRing | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Converts a Ring to a noncommutative ring
USAGE: S = toFreeAlgebraQuotient R
INPUTS: R : Ring
OUTPUTS: S : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
R = QQ[a,b,c,d, SkewCommutative=>{2,3}]
I = ideal(a*d-b*c)
S = R/I
S' = toFreeAlgebraQuotient(S)
ideal S'
```
SEEALSO: toCommRing | Key
toFreeAlgebraQuotient
(toFreeAlgebraQuotient,Ring)
Headline
Converts a Ring to a noncommutative ring
Usage
S = toFreeAlgebraQuotient R
Inputs
R : Ring
Outputs
S : FreeAlgebraQuotient
Description
Text
This function converts commutative rings and quotient... | ||
doc_c5489f44_(toFreeAlgebraQuotient,Ring) | doc | (toFreeAlgebraQuotient,Ring) | Converts a Ring to a noncommutative ring | S = toFreeAlgebraQuotient R | R = QQ[a,b,c,d, SkewCommutative=>{2,3}]
I = ideal(a*d-b*c)
S = R/I
S' = toFreeAlgebraQuotient(S)
ideal S' | toCommRing | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Converts a Ring to a noncommutative ring
USAGE: S = toFreeAlgebraQuotient R
INPUTS: R : Ring
OUTPUTS: S : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
R = QQ[a,b,c,d, SkewCommutative=>{2,3}]
I = ideal(a*d-b*c)
S = R/I
S' = toFreeAlgebraQuotient(S)
ideal S'
```
SEEALSO: toCommRing | Key
toFreeAlgebraQuotient
(toFreeAlgebraQuotient,Ring)
Headline
Converts a Ring to a noncommutative ring
Usage
S = toFreeAlgebraQuotient R
Inputs
R : Ring
Outputs
S : FreeAlgebraQuotient
Description
Text
This function converts commutative rings and quotient... | ||
doc_11614c72_skewPolynomialRing | doc | skewPolynomialRing | Defines a skew polynomial ring via a skewing matrix | B = skewPolynomialRing(R,M,L) | R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
M = matrix{{1,q,q},{q^4,1,1},{q^4,1,1}}
B = skewPolynomialRing(R,M,{x,y,z})
x*y == q^4*y*x
N = matrix{{1,1,1,1},{1,1,1,1},{1,1,1,1},{1,1,1,1}}
C = skewPolynomialRing(QQ,promote(N,QQ), {a,b,c,d})
isCommutative C
isCommutative B
Bop = opp... | oppositeRing | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Defines a skew polynomial ring via a skewing matrix
USAGE: B = skewPolynomialRing(R,M,L)
INPUTS: R : Ring
M : Matrix
L : List
OUTPUTS: B : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
M = matrix{{1,q,q},{q^4,1,1},{q^4,1,1}}
B = skewPolynomialRing(R... | Key
skewPolynomialRing
(skewPolynomialRing,Ring,Matrix,List)
Headline
Defines a skew polynomial ring via a skewing matrix
Usage
B = skewPolynomialRing(R,M,L)
Inputs
R : Ring
M : Matrix
L : List
Outputs
B : FreeAlgebraQuotient
Description
Text
... | ||
doc_11614c72_(skewPolynomialRing,Ring,Matrix,List) | doc | (skewPolynomialRing,Ring,Matrix,List) | Defines a skew polynomial ring via a skewing matrix | B = skewPolynomialRing(R,M,L) | R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
M = matrix{{1,q,q},{q^4,1,1},{q^4,1,1}}
B = skewPolynomialRing(R,M,{x,y,z})
x*y == q^4*y*x
N = matrix{{1,1,1,1},{1,1,1,1},{1,1,1,1},{1,1,1,1}}
C = skewPolynomialRing(QQ,promote(N,QQ), {a,b,c,d})
isCommutative C
isCommutative B
Bop = opp... | oppositeRing | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Defines a skew polynomial ring via a skewing matrix
USAGE: B = skewPolynomialRing(R,M,L)
INPUTS: R : Ring
M : Matrix
L : List
OUTPUTS: B : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
M = matrix{{1,q,q},{q^4,1,1},{q^4,1,1}}
B = skewPolynomialRing(R... | Key
skewPolynomialRing
(skewPolynomialRing,Ring,Matrix,List)
Headline
Defines a skew polynomial ring via a skewing matrix
Usage
B = skewPolynomialRing(R,M,L)
Inputs
R : Ring
M : Matrix
L : List
Outputs
B : FreeAlgebraQuotient
Description
Text
... | ||
doc_48548b10_(skewPolynomialRing,Ring,RingElement,List) | doc | (skewPolynomialRing,Ring,RingElement,List) | Defines a skew polynomial ring via a scaling factor | skewPolynomialRing(R,f,L) | R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
A = skewPolynomialRing(R,promote(2,R),{x,y,z,w})
x*y == 2*y*x
B = skewPolynomialRing(R,q,{x,y,z,w})
x*y == q*y*x
Bop = oppositeRing B
y*x == q*x*y
C = skewPolynomialRing(QQ,2_QQ, {x,y,z,w})
x*y == 2*y*x
... | oppositeRing
skewPolynomialRing | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Defines a skew polynomial ring via a scaling factor
USAGE: skewPolynomialRing(R,f,L)
INPUTS: R : Ring
f : RingElement
or an integer or a rational number
L : List
OUTPUTS: : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
A = skewPolynomialRing... | Key
(skewPolynomialRing,Ring,RingElement,List)
(skewPolynomialRing,Ring,QQ,List)
(skewPolynomialRing,Ring,ZZ,List)
Headline
Defines a skew polynomial ring via a scaling factor
Usage
skewPolynomialRing(R,f,L)
Inputs
R : Ring
f : RingElement
or an integer or a ... | ||
doc_48548b10_(skewPolynomialRing,Ring,QQ,List) | doc | (skewPolynomialRing,Ring,QQ,List) | Defines a skew polynomial ring via a scaling factor | skewPolynomialRing(R,f,L) | R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
A = skewPolynomialRing(R,promote(2,R),{x,y,z,w})
x*y == 2*y*x
B = skewPolynomialRing(R,q,{x,y,z,w})
x*y == q*y*x
Bop = oppositeRing B
y*x == q*x*y
C = skewPolynomialRing(QQ,2_QQ, {x,y,z,w})
x*y == 2*y*x
... | oppositeRing
skewPolynomialRing | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Defines a skew polynomial ring via a scaling factor
USAGE: skewPolynomialRing(R,f,L)
INPUTS: R : Ring
f : RingElement
or an integer or a rational number
L : List
OUTPUTS: : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
A = skewPolynomialRing... | Key
(skewPolynomialRing,Ring,RingElement,List)
(skewPolynomialRing,Ring,QQ,List)
(skewPolynomialRing,Ring,ZZ,List)
Headline
Defines a skew polynomial ring via a scaling factor
Usage
skewPolynomialRing(R,f,L)
Inputs
R : Ring
f : RingElement
or an integer or a ... | ||
doc_48548b10_(skewPolynomialRing,Ring,ZZ,List) | doc | (skewPolynomialRing,Ring,ZZ,List) | Defines a skew polynomial ring via a scaling factor | skewPolynomialRing(R,f,L) | R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
A = skewPolynomialRing(R,promote(2,R),{x,y,z,w})
x*y == 2*y*x
B = skewPolynomialRing(R,q,{x,y,z,w})
x*y == q*y*x
Bop = oppositeRing B
y*x == q*x*y
C = skewPolynomialRing(QQ,2_QQ, {x,y,z,w})
x*y == 2*y*x
... | oppositeRing
skewPolynomialRing | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Defines a skew polynomial ring via a scaling factor
USAGE: skewPolynomialRing(R,f,L)
INPUTS: R : Ring
f : RingElement
or an integer or a rational number
L : List
OUTPUTS: : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
A = skewPolynomialRing... | Key
(skewPolynomialRing,Ring,RingElement,List)
(skewPolynomialRing,Ring,QQ,List)
(skewPolynomialRing,Ring,ZZ,List)
Headline
Defines a skew polynomial ring via a scaling factor
Usage
skewPolynomialRing(R,f,L)
Inputs
R : Ring
f : RingElement
or an integer or a ... | ||
doc_e465ce09_oppositeRing | doc | oppositeRing | Creates the opposite ring of a noncommutative ring | Aop = oppositeRing A | R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
A = skewPolynomialRing(R,q,{x,y,z,w})
x*y == q*y*x
Aop = oppositeRing A
y*x == q*x*y | skewPolynomialRing | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates the opposite ring of a noncommutative ring
USAGE: Aop = oppositeRing A
INPUTS: A : FreeAlgebraQuotient
or @ TO FreeAlgebra @
OUTPUTS: Aop : FreeAlgebraQuotient
or @ TO FreeAlgebra @
EXAMPLE CODE:
```macaulay2
R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
A = skewPolynomialRing(R... | Key
oppositeRing
(oppositeRing,FreeAlgebra)
(oppositeRing,FreeAlgebraQuotient)
Headline
Creates the opposite ring of a noncommutative ring
Usage
Aop = oppositeRing A
Inputs
A : FreeAlgebraQuotient
or @ TO FreeAlgebra @
Outputs
... | ||
doc_e465ce09_(oppositeRing,FreeAlgebra) | doc | (oppositeRing,FreeAlgebra) | Creates the opposite ring of a noncommutative ring | Aop = oppositeRing A | R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
A = skewPolynomialRing(R,q,{x,y,z,w})
x*y == q*y*x
Aop = oppositeRing A
y*x == q*x*y | skewPolynomialRing | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates the opposite ring of a noncommutative ring
USAGE: Aop = oppositeRing A
INPUTS: A : FreeAlgebraQuotient
or @ TO FreeAlgebra @
OUTPUTS: Aop : FreeAlgebraQuotient
or @ TO FreeAlgebra @
EXAMPLE CODE:
```macaulay2
R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
A = skewPolynomialRing(R... | Key
oppositeRing
(oppositeRing,FreeAlgebra)
(oppositeRing,FreeAlgebraQuotient)
Headline
Creates the opposite ring of a noncommutative ring
Usage
Aop = oppositeRing A
Inputs
A : FreeAlgebraQuotient
or @ TO FreeAlgebra @
Outputs
... | ||
doc_e465ce09_(oppositeRing,FreeAlgebraQuotient) | doc | (oppositeRing,FreeAlgebraQuotient) | Creates the opposite ring of a noncommutative ring | Aop = oppositeRing A | R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
A = skewPolynomialRing(R,q,{x,y,z,w})
x*y == q*y*x
Aop = oppositeRing A
y*x == q*x*y | skewPolynomialRing | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Creates the opposite ring of a noncommutative ring
USAGE: Aop = oppositeRing A
INPUTS: A : FreeAlgebraQuotient
or @ TO FreeAlgebra @
OUTPUTS: Aop : FreeAlgebraQuotient
or @ TO FreeAlgebra @
EXAMPLE CODE:
```macaulay2
R = QQ[q]/ideal{q^4+q^3+q^2+q+1}
A = skewPolynomialRing(R... | Key
oppositeRing
(oppositeRing,FreeAlgebra)
(oppositeRing,FreeAlgebraQuotient)
Headline
Creates the opposite ring of a noncommutative ring
Usage
Aop = oppositeRing A
Inputs
A : FreeAlgebraQuotient
or @ TO FreeAlgebra @
Outputs
... | ||
doc_908084e2_ncHilbertSeries | doc | ncHilbertSeries | Computes the Hilbert series of a noncommutative ring | hseries = ncHilbertSeries(A) | A = QQ<|x,y,z|>
ncHilbertSeries(A,Order=>10)
A = QQ<|x,y,z,Degrees=>{1,2,3}|>
ncHilbertSeries(A,Order=>10)
B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
ncHilbertSeries(B,Order=>10) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the Hilbert series of a noncommutative ring
USAGE: hseries = ncHilbertSeries(A)
INPUTS: A : FreeAlgebraQuotient
or @ TO FreeAlgebra @
OUTPUTS: : Expression
or @ TO RingElement @
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
ncHilbertSeries(A,Order=>10)
A = QQ<|x,y,z,Degrees=>{1... | Key
ncHilbertSeries
(ncHilbertSeries, FreeAlgebraQuotient)
(ncHilbertSeries, FreeAlgebra)
[ncHilbertSeries, Order]
Headline
Computes the Hilbert series of a noncommutative ring
Usage
hseries = ncHilbertSeries(A)
Inputs
A : FreeAlgebraQuotient
or @ TO FreeAlgebra... | |||
doc_908084e2_(ncHilbertSeries,_FreeAlgebraQuotient) | doc | (ncHilbertSeries, FreeAlgebraQuotient) | Computes the Hilbert series of a noncommutative ring | hseries = ncHilbertSeries(A) | A = QQ<|x,y,z|>
ncHilbertSeries(A,Order=>10)
A = QQ<|x,y,z,Degrees=>{1,2,3}|>
ncHilbertSeries(A,Order=>10)
B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
ncHilbertSeries(B,Order=>10) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the Hilbert series of a noncommutative ring
USAGE: hseries = ncHilbertSeries(A)
INPUTS: A : FreeAlgebraQuotient
or @ TO FreeAlgebra @
OUTPUTS: : Expression
or @ TO RingElement @
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
ncHilbertSeries(A,Order=>10)
A = QQ<|x,y,z,Degrees=>{1... | Key
ncHilbertSeries
(ncHilbertSeries, FreeAlgebraQuotient)
(ncHilbertSeries, FreeAlgebra)
[ncHilbertSeries, Order]
Headline
Computes the Hilbert series of a noncommutative ring
Usage
hseries = ncHilbertSeries(A)
Inputs
A : FreeAlgebraQuotient
or @ TO FreeAlgebra... | |||
doc_908084e2_(ncHilbertSeries,_FreeAlgebra) | doc | (ncHilbertSeries, FreeAlgebra) | Computes the Hilbert series of a noncommutative ring | hseries = ncHilbertSeries(A) | A = QQ<|x,y,z|>
ncHilbertSeries(A,Order=>10)
A = QQ<|x,y,z,Degrees=>{1,2,3}|>
ncHilbertSeries(A,Order=>10)
B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
ncHilbertSeries(B,Order=>10) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the Hilbert series of a noncommutative ring
USAGE: hseries = ncHilbertSeries(A)
INPUTS: A : FreeAlgebraQuotient
or @ TO FreeAlgebra @
OUTPUTS: : Expression
or @ TO RingElement @
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
ncHilbertSeries(A,Order=>10)
A = QQ<|x,y,z,Degrees=>{1... | Key
ncHilbertSeries
(ncHilbertSeries, FreeAlgebraQuotient)
(ncHilbertSeries, FreeAlgebra)
[ncHilbertSeries, Order]
Headline
Computes the Hilbert series of a noncommutative ring
Usage
hseries = ncHilbertSeries(A)
Inputs
A : FreeAlgebraQuotient
or @ TO FreeAlgebra... | |||
doc_908084e2_[ncHilbertSeries,_Order] | doc | [ncHilbertSeries, Order] | Computes the Hilbert series of a noncommutative ring | hseries = ncHilbertSeries(A) | A = QQ<|x,y,z|>
ncHilbertSeries(A,Order=>10)
A = QQ<|x,y,z,Degrees=>{1,2,3}|>
ncHilbertSeries(A,Order=>10)
B = threeDimSklyanin(QQ,{1,1,-1},{x,y,z})
ncHilbertSeries(B,Order=>10) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Computes the Hilbert series of a noncommutative ring
USAGE: hseries = ncHilbertSeries(A)
INPUTS: A : FreeAlgebraQuotient
or @ TO FreeAlgebra @
OUTPUTS: : Expression
or @ TO RingElement @
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
ncHilbertSeries(A,Order=>10)
A = QQ<|x,y,z,Degrees=>{1... | Key
ncHilbertSeries
(ncHilbertSeries, FreeAlgebraQuotient)
(ncHilbertSeries, FreeAlgebra)
[ncHilbertSeries, Order]
Headline
Computes the Hilbert series of a noncommutative ring
Usage
hseries = ncHilbertSeries(A)
Inputs
A : FreeAlgebraQuotient
or @ TO FreeAlgebra... | |||
doc_ac84ea25_endomorphismRingIdeal | doc | endomorphismRingIdeal | Find the relations of an endomorphism ring | I = endomorphismRingIdeal(M,X) | maps = I.cache#"EndomorphismRingIdealGens"
assert(maps_0*maps_2 == maps_3) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Find the relations of an endomorphism ring
USAGE: I = endomorphismRingIdeal(M,X)
INPUTS: M : Module
X : Symbol
the base name for the indexed variables serving as generators for the output ring
OUTPUTS: I : Ideal
in a FreeAlgebra with variables with base name X
EXAMPLE CODE:
```mac... | Key
endomorphismRingIdeal
(endomorphismRingIdeal,Module,Symbol)
Headline
Find the relations of an endomorphism ring
Usage
I = endomorphismRingIdeal(M,X)
Inputs
M : Module
X : Symbol
the base name for the indexed variables serving as generators for the output ring
... | |||
doc_ac84ea25_(endomorphismRingIdeal,Module,Symbol) | doc | (endomorphismRingIdeal,Module,Symbol) | Find the relations of an endomorphism ring | I = endomorphismRingIdeal(M,X) | maps = I.cache#"EndomorphismRingIdealGens"
assert(maps_0*maps_2 == maps_3) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Find the relations of an endomorphism ring
USAGE: I = endomorphismRingIdeal(M,X)
INPUTS: M : Module
X : Symbol
the base name for the indexed variables serving as generators for the output ring
OUTPUTS: I : Ideal
in a FreeAlgebra with variables with base name X
EXAMPLE CODE:
```mac... | Key
endomorphismRingIdeal
(endomorphismRingIdeal,Module,Symbol)
Headline
Find the relations of an endomorphism ring
Usage
I = endomorphismRingIdeal(M,X)
Inputs
M : Module
X : Symbol
the base name for the indexed variables serving as generators for the output ring
... | |||
doc_64b868d8_extAlgebra | doc | extAlgebra | Compute the Ext algebra of a ring | extAlgebra(R,z) | ER.cache#"extMaps"#(z_4) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the Ext algebra of a ring
USAGE: extAlgebra(R,z)
INPUTS: R : Ring
z : Symbol
OUTPUTS: : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
ER.cache#"extMaps"#(z_4)
``` | Key
extAlgebra
(extAlgebra,Ring,Symbol)
[extAlgebra,DegreeLimit]
Headline
Compute the Ext algebra of a ring
Usage
extAlgebra(R,z)
Inputs
R : Ring
z : Symbol
Outputs
: FreeAlgebraQuotient
Description
Text
This command uses the functions @ TO yonedaMap ... | |||
doc_64b868d8_(extAlgebra,Ring,Symbol) | doc | (extAlgebra,Ring,Symbol) | Compute the Ext algebra of a ring | extAlgebra(R,z) | ER.cache#"extMaps"#(z_4) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the Ext algebra of a ring
USAGE: extAlgebra(R,z)
INPUTS: R : Ring
z : Symbol
OUTPUTS: : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
ER.cache#"extMaps"#(z_4)
``` | Key
extAlgebra
(extAlgebra,Ring,Symbol)
[extAlgebra,DegreeLimit]
Headline
Compute the Ext algebra of a ring
Usage
extAlgebra(R,z)
Inputs
R : Ring
z : Symbol
Outputs
: FreeAlgebraQuotient
Description
Text
This command uses the functions @ TO yonedaMap ... | |||
doc_64b868d8_[extAlgebra,DegreeLimit] | doc | [extAlgebra,DegreeLimit] | Compute the Ext algebra of a ring | extAlgebra(R,z) | ER.cache#"extMaps"#(z_4) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the Ext algebra of a ring
USAGE: extAlgebra(R,z)
INPUTS: R : Ring
z : Symbol
OUTPUTS: : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
ER.cache#"extMaps"#(z_4)
``` | Key
extAlgebra
(extAlgebra,Ring,Symbol)
[extAlgebra,DegreeLimit]
Headline
Compute the Ext algebra of a ring
Usage
extAlgebra(R,z)
Inputs
R : Ring
z : Symbol
Outputs
: FreeAlgebraQuotient
Description
Text
This command uses the functions @ TO yonedaMap ... | |||
doc_0d7635c3_NCReductionTwoSided | doc | NCReductionTwoSided | Reduces the entries of an Matrix with respect to an ideal | L = NCReductionTwoSided(M,I) | A = QQ<|x,y,z|>
f = y*z + z*y - x^2
g = x*z + z*x - y^2
h = z^2 - x*y - y*x
I = ideal {f,g,h}
Igb = NCGB(I,10)
NCReductionTwoSided(x^4,I)
NCReductionTwoSided(x^4,Igb) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Reduces the entries of an Matrix with respect to an ideal
USAGE: L = NCReductionTwoSided(M,I)
INPUTS: M : Matrix
I : Ideal
OUTPUTS: L : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
f = y*z + z*y - x^2
g = x*z + z*x - y^2
h = z^2 - x*y - y*x
I = ideal {f,g,h}
Igb = NCGB(I,10)
NCReduct... | Key
NCReductionTwoSided
(NCReductionTwoSided, RingElement, List)
(NCReductionTwoSided, RingElement, Ideal)
(NCReductionTwoSided, RingElement, Matrix)
(NCReductionTwoSided, Matrix, List)
(NCReductionTwoSided, Matrix, Ideal)
(NCReductionTwoSided, Matrix, Matrix)
Headline
... | |||
doc_0d7635c3_(NCReductionTwoSided,_RingElement,_List) | doc | (NCReductionTwoSided, RingElement, List) | Reduces the entries of an Matrix with respect to an ideal | L = NCReductionTwoSided(M,I) | A = QQ<|x,y,z|>
f = y*z + z*y - x^2
g = x*z + z*x - y^2
h = z^2 - x*y - y*x
I = ideal {f,g,h}
Igb = NCGB(I,10)
NCReductionTwoSided(x^4,I)
NCReductionTwoSided(x^4,Igb) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Reduces the entries of an Matrix with respect to an ideal
USAGE: L = NCReductionTwoSided(M,I)
INPUTS: M : Matrix
I : Ideal
OUTPUTS: L : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
f = y*z + z*y - x^2
g = x*z + z*x - y^2
h = z^2 - x*y - y*x
I = ideal {f,g,h}
Igb = NCGB(I,10)
NCReduct... | Key
NCReductionTwoSided
(NCReductionTwoSided, RingElement, List)
(NCReductionTwoSided, RingElement, Ideal)
(NCReductionTwoSided, RingElement, Matrix)
(NCReductionTwoSided, Matrix, List)
(NCReductionTwoSided, Matrix, Ideal)
(NCReductionTwoSided, Matrix, Matrix)
Headline
... | |||
doc_0d7635c3_(NCReductionTwoSided,_RingElement,_Ideal) | doc | (NCReductionTwoSided, RingElement, Ideal) | Reduces the entries of an Matrix with respect to an ideal | L = NCReductionTwoSided(M,I) | A = QQ<|x,y,z|>
f = y*z + z*y - x^2
g = x*z + z*x - y^2
h = z^2 - x*y - y*x
I = ideal {f,g,h}
Igb = NCGB(I,10)
NCReductionTwoSided(x^4,I)
NCReductionTwoSided(x^4,Igb) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Reduces the entries of an Matrix with respect to an ideal
USAGE: L = NCReductionTwoSided(M,I)
INPUTS: M : Matrix
I : Ideal
OUTPUTS: L : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
f = y*z + z*y - x^2
g = x*z + z*x - y^2
h = z^2 - x*y - y*x
I = ideal {f,g,h}
Igb = NCGB(I,10)
NCReduct... | Key
NCReductionTwoSided
(NCReductionTwoSided, RingElement, List)
(NCReductionTwoSided, RingElement, Ideal)
(NCReductionTwoSided, RingElement, Matrix)
(NCReductionTwoSided, Matrix, List)
(NCReductionTwoSided, Matrix, Ideal)
(NCReductionTwoSided, Matrix, Matrix)
Headline
... | |||
doc_0d7635c3_(NCReductionTwoSided,_RingElement,_Matrix) | doc | (NCReductionTwoSided, RingElement, Matrix) | Reduces the entries of an Matrix with respect to an ideal | L = NCReductionTwoSided(M,I) | A = QQ<|x,y,z|>
f = y*z + z*y - x^2
g = x*z + z*x - y^2
h = z^2 - x*y - y*x
I = ideal {f,g,h}
Igb = NCGB(I,10)
NCReductionTwoSided(x^4,I)
NCReductionTwoSided(x^4,Igb) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Reduces the entries of an Matrix with respect to an ideal
USAGE: L = NCReductionTwoSided(M,I)
INPUTS: M : Matrix
I : Ideal
OUTPUTS: L : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
f = y*z + z*y - x^2
g = x*z + z*x - y^2
h = z^2 - x*y - y*x
I = ideal {f,g,h}
Igb = NCGB(I,10)
NCReduct... | Key
NCReductionTwoSided
(NCReductionTwoSided, RingElement, List)
(NCReductionTwoSided, RingElement, Ideal)
(NCReductionTwoSided, RingElement, Matrix)
(NCReductionTwoSided, Matrix, List)
(NCReductionTwoSided, Matrix, Ideal)
(NCReductionTwoSided, Matrix, Matrix)
Headline
... | |||
doc_0d7635c3_(NCReductionTwoSided,_Matrix,_List) | doc | (NCReductionTwoSided, Matrix, List) | Reduces the entries of an Matrix with respect to an ideal | L = NCReductionTwoSided(M,I) | A = QQ<|x,y,z|>
f = y*z + z*y - x^2
g = x*z + z*x - y^2
h = z^2 - x*y - y*x
I = ideal {f,g,h}
Igb = NCGB(I,10)
NCReductionTwoSided(x^4,I)
NCReductionTwoSided(x^4,Igb) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Reduces the entries of an Matrix with respect to an ideal
USAGE: L = NCReductionTwoSided(M,I)
INPUTS: M : Matrix
I : Ideal
OUTPUTS: L : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
f = y*z + z*y - x^2
g = x*z + z*x - y^2
h = z^2 - x*y - y*x
I = ideal {f,g,h}
Igb = NCGB(I,10)
NCReduct... | Key
NCReductionTwoSided
(NCReductionTwoSided, RingElement, List)
(NCReductionTwoSided, RingElement, Ideal)
(NCReductionTwoSided, RingElement, Matrix)
(NCReductionTwoSided, Matrix, List)
(NCReductionTwoSided, Matrix, Ideal)
(NCReductionTwoSided, Matrix, Matrix)
Headline
... | |||
doc_0d7635c3_(NCReductionTwoSided,_Matrix,_Ideal) | doc | (NCReductionTwoSided, Matrix, Ideal) | Reduces the entries of an Matrix with respect to an ideal | L = NCReductionTwoSided(M,I) | A = QQ<|x,y,z|>
f = y*z + z*y - x^2
g = x*z + z*x - y^2
h = z^2 - x*y - y*x
I = ideal {f,g,h}
Igb = NCGB(I,10)
NCReductionTwoSided(x^4,I)
NCReductionTwoSided(x^4,Igb) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Reduces the entries of an Matrix with respect to an ideal
USAGE: L = NCReductionTwoSided(M,I)
INPUTS: M : Matrix
I : Ideal
OUTPUTS: L : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
f = y*z + z*y - x^2
g = x*z + z*x - y^2
h = z^2 - x*y - y*x
I = ideal {f,g,h}
Igb = NCGB(I,10)
NCReduct... | Key
NCReductionTwoSided
(NCReductionTwoSided, RingElement, List)
(NCReductionTwoSided, RingElement, Ideal)
(NCReductionTwoSided, RingElement, Matrix)
(NCReductionTwoSided, Matrix, List)
(NCReductionTwoSided, Matrix, Ideal)
(NCReductionTwoSided, Matrix, Matrix)
Headline
... | |||
doc_0d7635c3_(NCReductionTwoSided,_Matrix,_Matrix) | doc | (NCReductionTwoSided, Matrix, Matrix) | Reduces the entries of an Matrix with respect to an ideal | L = NCReductionTwoSided(M,I) | A = QQ<|x,y,z|>
f = y*z + z*y - x^2
g = x*z + z*x - y^2
h = z^2 - x*y - y*x
I = ideal {f,g,h}
Igb = NCGB(I,10)
NCReductionTwoSided(x^4,I)
NCReductionTwoSided(x^4,Igb) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Reduces the entries of an Matrix with respect to an ideal
USAGE: L = NCReductionTwoSided(M,I)
INPUTS: M : Matrix
I : Ideal
OUTPUTS: L : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
f = y*z + z*y - x^2
g = x*z + z*x - y^2
h = z^2 - x*y - y*x
I = ideal {f,g,h}
Igb = NCGB(I,10)
NCReduct... | Key
NCReductionTwoSided
(NCReductionTwoSided, RingElement, List)
(NCReductionTwoSided, RingElement, Ideal)
(NCReductionTwoSided, RingElement, Matrix)
(NCReductionTwoSided, Matrix, List)
(NCReductionTwoSided, Matrix, Ideal)
(NCReductionTwoSided, Matrix, Matrix)
Headline
... | |||
doc_5aedfefb_leftQuadraticMatrix | doc | leftQuadraticMatrix | Factors the quadratic ideal on the left or on the right. | M = leftQuadraticMatrix I | S = R/I
(lQS,dS) = (sub(lQ,S),sub(d,S));
(rQS,eS) = (sub(rQ,S),sub(e,S));
ncMatrixMult(dS,rQS)
ncMatrixMult(lQS,eS) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Factors the quadratic ideal on the left or on the right.
USAGE: M = leftQuadraticMatrix I
INPUTS: I : Ideal
or @ TO List @
OUTPUTS: M : Matrix
EXAMPLE CODE:
```macaulay2
S = R/I
(lQS,dS) = (sub(lQ,S),sub(d,S));
(rQS,eS) = (sub(rQ,S),sub(e,S));
ncMatrixMult(dS,rQS)
ncMatrixMult(lQS,eS)
``` | Key
leftQuadraticMatrix
(leftQuadraticMatrix,List)
(leftQuadraticMatrix,Ideal)
rightQuadraticMatrix
(rightQuadraticMatrix,List)
(rightQuadraticMatrix,Ideal)
Headline
Factors the quadratic ideal on the left or on the right.
Usage
M = leftQuadraticMatrix I
Inputs
I : ... | |||
doc_5aedfefb_(leftQuadraticMatrix,List) | doc | (leftQuadraticMatrix,List) | Factors the quadratic ideal on the left or on the right. | M = leftQuadraticMatrix I | S = R/I
(lQS,dS) = (sub(lQ,S),sub(d,S));
(rQS,eS) = (sub(rQ,S),sub(e,S));
ncMatrixMult(dS,rQS)
ncMatrixMult(lQS,eS) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Factors the quadratic ideal on the left or on the right.
USAGE: M = leftQuadraticMatrix I
INPUTS: I : Ideal
or @ TO List @
OUTPUTS: M : Matrix
EXAMPLE CODE:
```macaulay2
S = R/I
(lQS,dS) = (sub(lQ,S),sub(d,S));
(rQS,eS) = (sub(rQ,S),sub(e,S));
ncMatrixMult(dS,rQS)
ncMatrixMult(lQS,eS)
``` | Key
leftQuadraticMatrix
(leftQuadraticMatrix,List)
(leftQuadraticMatrix,Ideal)
rightQuadraticMatrix
(rightQuadraticMatrix,List)
(rightQuadraticMatrix,Ideal)
Headline
Factors the quadratic ideal on the left or on the right.
Usage
M = leftQuadraticMatrix I
Inputs
I : ... | |||
doc_5aedfefb_(leftQuadraticMatrix,Ideal) | doc | (leftQuadraticMatrix,Ideal) | Factors the quadratic ideal on the left or on the right. | M = leftQuadraticMatrix I | S = R/I
(lQS,dS) = (sub(lQ,S),sub(d,S));
(rQS,eS) = (sub(rQ,S),sub(e,S));
ncMatrixMult(dS,rQS)
ncMatrixMult(lQS,eS) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Factors the quadratic ideal on the left or on the right.
USAGE: M = leftQuadraticMatrix I
INPUTS: I : Ideal
or @ TO List @
OUTPUTS: M : Matrix
EXAMPLE CODE:
```macaulay2
S = R/I
(lQS,dS) = (sub(lQ,S),sub(d,S));
(rQS,eS) = (sub(rQ,S),sub(e,S));
ncMatrixMult(dS,rQS)
ncMatrixMult(lQS,eS)
``` | Key
leftQuadraticMatrix
(leftQuadraticMatrix,List)
(leftQuadraticMatrix,Ideal)
rightQuadraticMatrix
(rightQuadraticMatrix,List)
(rightQuadraticMatrix,Ideal)
Headline
Factors the quadratic ideal on the left or on the right.
Usage
M = leftQuadraticMatrix I
Inputs
I : ... | |||
doc_5aedfefb_rightQuadraticMatrix | doc | rightQuadraticMatrix | Factors the quadratic ideal on the left or on the right. | M = leftQuadraticMatrix I | S = R/I
(lQS,dS) = (sub(lQ,S),sub(d,S));
(rQS,eS) = (sub(rQ,S),sub(e,S));
ncMatrixMult(dS,rQS)
ncMatrixMult(lQS,eS) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Factors the quadratic ideal on the left or on the right.
USAGE: M = leftQuadraticMatrix I
INPUTS: I : Ideal
or @ TO List @
OUTPUTS: M : Matrix
EXAMPLE CODE:
```macaulay2
S = R/I
(lQS,dS) = (sub(lQ,S),sub(d,S));
(rQS,eS) = (sub(rQ,S),sub(e,S));
ncMatrixMult(dS,rQS)
ncMatrixMult(lQS,eS)
``` | Key
leftQuadraticMatrix
(leftQuadraticMatrix,List)
(leftQuadraticMatrix,Ideal)
rightQuadraticMatrix
(rightQuadraticMatrix,List)
(rightQuadraticMatrix,Ideal)
Headline
Factors the quadratic ideal on the left or on the right.
Usage
M = leftQuadraticMatrix I
Inputs
I : ... | |||
doc_5aedfefb_(rightQuadraticMatrix,List) | doc | (rightQuadraticMatrix,List) | Factors the quadratic ideal on the left or on the right. | M = leftQuadraticMatrix I | S = R/I
(lQS,dS) = (sub(lQ,S),sub(d,S));
(rQS,eS) = (sub(rQ,S),sub(e,S));
ncMatrixMult(dS,rQS)
ncMatrixMult(lQS,eS) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Factors the quadratic ideal on the left or on the right.
USAGE: M = leftQuadraticMatrix I
INPUTS: I : Ideal
or @ TO List @
OUTPUTS: M : Matrix
EXAMPLE CODE:
```macaulay2
S = R/I
(lQS,dS) = (sub(lQ,S),sub(d,S));
(rQS,eS) = (sub(rQ,S),sub(e,S));
ncMatrixMult(dS,rQS)
ncMatrixMult(lQS,eS)
``` | Key
leftQuadraticMatrix
(leftQuadraticMatrix,List)
(leftQuadraticMatrix,Ideal)
rightQuadraticMatrix
(rightQuadraticMatrix,List)
(rightQuadraticMatrix,Ideal)
Headline
Factors the quadratic ideal on the left or on the right.
Usage
M = leftQuadraticMatrix I
Inputs
I : ... | |||
doc_5aedfefb_(rightQuadraticMatrix,Ideal) | doc | (rightQuadraticMatrix,Ideal) | Factors the quadratic ideal on the left or on the right. | M = leftQuadraticMatrix I | S = R/I
(lQS,dS) = (sub(lQ,S),sub(d,S));
(rQS,eS) = (sub(rQ,S),sub(e,S));
ncMatrixMult(dS,rQS)
ncMatrixMult(lQS,eS) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Factors the quadratic ideal on the left or on the right.
USAGE: M = leftQuadraticMatrix I
INPUTS: I : Ideal
or @ TO List @
OUTPUTS: M : Matrix
EXAMPLE CODE:
```macaulay2
S = R/I
(lQS,dS) = (sub(lQ,S),sub(d,S));
(rQS,eS) = (sub(rQ,S),sub(e,S));
ncMatrixMult(dS,rQS)
ncMatrixMult(lQS,eS)
``` | Key
leftQuadraticMatrix
(leftQuadraticMatrix,List)
(leftQuadraticMatrix,Ideal)
rightQuadraticMatrix
(rightQuadraticMatrix,List)
(rightQuadraticMatrix,Ideal)
Headline
Factors the quadratic ideal on the left or on the right.
Usage
M = leftQuadraticMatrix I
Inputs
I : ... | |||
doc_72bab294_ncMatrixMult | doc | ncMatrixMult | Correctly multiplies matrices from noncommutative rings. | L = ncMatrixMult(M,N) | A = QQ<|x,y|>
M = matrix {{x}}
N = matrix {{y}}
M*N
assert(ncMatrixMult(M,N) == matrix {{x*y}}) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Correctly multiplies matrices from noncommutative rings.
USAGE: L = ncMatrixMult(M,N)
INPUTS: M : Matrix
N : Matrix
OUTPUTS: L : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
M = matrix {{x}}
N = matrix {{y}}
M*N
assert(ncMatrixMult(M,N) == matrix {{x*y}})
``` | Key
ncMatrixMult
(ncMatrixMult, Matrix, Matrix)
Headline
Correctly multiplies matrices from noncommutative rings.
Usage
L = ncMatrixMult(M,N)
Inputs
M : Matrix
N : Matrix
Outputs
L : Matrix
Description
Text
This function is provided as a temporary band-aid... | |||
doc_72bab294_(ncMatrixMult,_Matrix,_Matrix) | doc | (ncMatrixMult, Matrix, Matrix) | Correctly multiplies matrices from noncommutative rings. | L = ncMatrixMult(M,N) | A = QQ<|x,y|>
M = matrix {{x}}
N = matrix {{y}}
M*N
assert(ncMatrixMult(M,N) == matrix {{x*y}}) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Correctly multiplies matrices from noncommutative rings.
USAGE: L = ncMatrixMult(M,N)
INPUTS: M : Matrix
N : Matrix
OUTPUTS: L : Matrix
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
M = matrix {{x}}
N = matrix {{y}}
M*N
assert(ncMatrixMult(M,N) == matrix {{x*y}})
``` | Key
ncMatrixMult
(ncMatrixMult, Matrix, Matrix)
Headline
Correctly multiplies matrices from noncommutative rings.
Usage
L = ncMatrixMult(M,N)
Inputs
M : Matrix
N : Matrix
Outputs
L : Matrix
Description
Text
This function is provided as a temporary band-aid... | |||
doc_d387b373_freeAlgebra | doc | freeAlgebra | Create a FreeAlgebra | A = freeAlgebra(R,xs) | B = freeAlgebra(QQ,{x,y,a,b,c,Weights=>{1,1,0,0,0},Degrees=>{1,1,3,3,2}})
I = ideal {a - x*y*x, b - y*x*y, c - x*y}
Igb = NCGB(I,10) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Create a FreeAlgebra
USAGE: A = freeAlgebra(R,xs)
INPUTS: R : Ring
xs : BasicList
containing the variables, and any options
OUTPUTS: A : FreeAlgebra
EXAMPLE CODE:
```macaulay2
B = freeAlgebra(QQ,{x,y,a,b,c,Weights=>{1,1,0,0,0},Degrees=>{1,1,3,3,2}})
I = ideal {a - x*y*x, b - y*x*y, c... | Key
freeAlgebra
(freeAlgebra,Ring,BasicList)
UseVariables
Headline
Create a FreeAlgebra
Usage
A = freeAlgebra(R,xs)
Inputs
R : Ring
xs : BasicList
containing the variables, and any options
Outputs
A : FreeAlgebra
Description
Text
This functio... | |||
doc_d387b373_(freeAlgebra,Ring,BasicList) | doc | (freeAlgebra,Ring,BasicList) | Create a FreeAlgebra | A = freeAlgebra(R,xs) | B = freeAlgebra(QQ,{x,y,a,b,c,Weights=>{1,1,0,0,0},Degrees=>{1,1,3,3,2}})
I = ideal {a - x*y*x, b - y*x*y, c - x*y}
Igb = NCGB(I,10) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Create a FreeAlgebra
USAGE: A = freeAlgebra(R,xs)
INPUTS: R : Ring
xs : BasicList
containing the variables, and any options
OUTPUTS: A : FreeAlgebra
EXAMPLE CODE:
```macaulay2
B = freeAlgebra(QQ,{x,y,a,b,c,Weights=>{1,1,0,0,0},Degrees=>{1,1,3,3,2}})
I = ideal {a - x*y*x, b - y*x*y, c... | Key
freeAlgebra
(freeAlgebra,Ring,BasicList)
UseVariables
Headline
Create a FreeAlgebra
Usage
A = freeAlgebra(R,xs)
Inputs
R : Ring
xs : BasicList
containing the variables, and any options
Outputs
A : FreeAlgebra
Description
Text
This functio... | |||
doc_d387b373_UseVariables | doc | UseVariables | Create a FreeAlgebra | A = freeAlgebra(R,xs) | B = freeAlgebra(QQ,{x,y,a,b,c,Weights=>{1,1,0,0,0},Degrees=>{1,1,3,3,2}})
I = ideal {a - x*y*x, b - y*x*y, c - x*y}
Igb = NCGB(I,10) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Create a FreeAlgebra
USAGE: A = freeAlgebra(R,xs)
INPUTS: R : Ring
xs : BasicList
containing the variables, and any options
OUTPUTS: A : FreeAlgebra
EXAMPLE CODE:
```macaulay2
B = freeAlgebra(QQ,{x,y,a,b,c,Weights=>{1,1,0,0,0},Degrees=>{1,1,3,3,2}})
I = ideal {a - x*y*x, b - y*x*y, c... | Key
freeAlgebra
(freeAlgebra,Ring,BasicList)
UseVariables
Headline
Create a FreeAlgebra
Usage
A = freeAlgebra(R,xs)
Inputs
R : Ring
xs : BasicList
containing the variables, and any options
Outputs
A : FreeAlgebra
Description
Text
This functio... | |||
doc_857b3988_ncGraphIdeal | doc | ncGraphIdeal | Compute the graph ideal of a ring map between noncommutative rings. | I = ncGraphIdeal f | A = QQ<|a,b,c|>
B = QQ<|x,y|>
f = map(B,A,{x*y*x,y*x*y,x*y})
I = ncGraphIdeal f
Igb = NCGB(I,10) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the graph ideal of a ring map between noncommutative rings.
USAGE: I = ncGraphIdeal f
INPUTS: f : RingMap
OUTPUTS: I : Ideal
EXAMPLE CODE:
```macaulay2
A = QQ<|a,b,c|>
B = QQ<|x,y|>
f = map(B,A,{x*y*x,y*x*y,x*y})
I = ncGraphIdeal f
Igb = NCGB(I,10)
``` | Key
ncGraphIdeal
(ncGraphIdeal,RingMap)
Headline
Compute the graph ideal of a ring map between noncommutative rings.
Usage
I = ncGraphIdeal f
Inputs
f : RingMap
Outputs
I : Ideal
Description
Text
This function creates the graph ideal of a ring map between nonco... | |||
doc_857b3988_(ncGraphIdeal,RingMap) | doc | (ncGraphIdeal,RingMap) | Compute the graph ideal of a ring map between noncommutative rings. | I = ncGraphIdeal f | A = QQ<|a,b,c|>
B = QQ<|x,y|>
f = map(B,A,{x*y*x,y*x*y,x*y})
I = ncGraphIdeal f
Igb = NCGB(I,10) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the graph ideal of a ring map between noncommutative rings.
USAGE: I = ncGraphIdeal f
INPUTS: f : RingMap
OUTPUTS: I : Ideal
EXAMPLE CODE:
```macaulay2
A = QQ<|a,b,c|>
B = QQ<|x,y|>
f = map(B,A,{x*y*x,y*x*y,x*y})
I = ncGraphIdeal f
Igb = NCGB(I,10)
``` | Key
ncGraphIdeal
(ncGraphIdeal,RingMap)
Headline
Compute the graph ideal of a ring map between noncommutative rings.
Usage
I = ncGraphIdeal f
Inputs
f : RingMap
Outputs
I : Ideal
Description
Text
This function creates the graph ideal of a ring map between nonco... | |||
doc_79d91653_ncKernel | doc | ncKernel | Compute the graph ideal of a ring map between noncommutative rings. | I = ncKernel f | A = QQ<|a,b,c|>
B = QQ<|x,y|>
f = map(B,A,{x*y*x,y*x*y,x*y})
K = ncKernel f | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the graph ideal of a ring map between noncommutative rings.
USAGE: I = ncKernel f
INPUTS: f : RingMap
OUTPUTS: I : Ideal
EXAMPLE CODE:
```macaulay2
A = QQ<|a,b,c|>
B = QQ<|x,y|>
f = map(B,A,{x*y*x,y*x*y,x*y})
K = ncKernel f
``` | Key
ncKernel
(ncKernel,RingMap)
[ncKernel,DegreeLimit]
[ncKernel,Strategy]
Headline
Compute the graph ideal of a ring map between noncommutative rings.
Usage
I = ncKernel f
Inputs
f : RingMap
Outputs
I : Ideal
Description
Text
This function comp... | |||
doc_79d91653_(ncKernel,RingMap) | doc | (ncKernel,RingMap) | Compute the graph ideal of a ring map between noncommutative rings. | I = ncKernel f | A = QQ<|a,b,c|>
B = QQ<|x,y|>
f = map(B,A,{x*y*x,y*x*y,x*y})
K = ncKernel f | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the graph ideal of a ring map between noncommutative rings.
USAGE: I = ncKernel f
INPUTS: f : RingMap
OUTPUTS: I : Ideal
EXAMPLE CODE:
```macaulay2
A = QQ<|a,b,c|>
B = QQ<|x,y|>
f = map(B,A,{x*y*x,y*x*y,x*y})
K = ncKernel f
``` | Key
ncKernel
(ncKernel,RingMap)
[ncKernel,DegreeLimit]
[ncKernel,Strategy]
Headline
Compute the graph ideal of a ring map between noncommutative rings.
Usage
I = ncKernel f
Inputs
f : RingMap
Outputs
I : Ideal
Description
Text
This function comp... | |||
doc_79d91653_[ncKernel,DegreeLimit] | doc | [ncKernel,DegreeLimit] | Compute the graph ideal of a ring map between noncommutative rings. | I = ncKernel f | A = QQ<|a,b,c|>
B = QQ<|x,y|>
f = map(B,A,{x*y*x,y*x*y,x*y})
K = ncKernel f | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the graph ideal of a ring map between noncommutative rings.
USAGE: I = ncKernel f
INPUTS: f : RingMap
OUTPUTS: I : Ideal
EXAMPLE CODE:
```macaulay2
A = QQ<|a,b,c|>
B = QQ<|x,y|>
f = map(B,A,{x*y*x,y*x*y,x*y})
K = ncKernel f
``` | Key
ncKernel
(ncKernel,RingMap)
[ncKernel,DegreeLimit]
[ncKernel,Strategy]
Headline
Compute the graph ideal of a ring map between noncommutative rings.
Usage
I = ncKernel f
Inputs
f : RingMap
Outputs
I : Ideal
Description
Text
This function comp... | |||
doc_79d91653_[ncKernel,Strategy] | doc | [ncKernel,Strategy] | Compute the graph ideal of a ring map between noncommutative rings. | I = ncKernel f | A = QQ<|a,b,c|>
B = QQ<|x,y|>
f = map(B,A,{x*y*x,y*x*y,x*y})
K = ncKernel f | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the graph ideal of a ring map between noncommutative rings.
USAGE: I = ncKernel f
INPUTS: f : RingMap
OUTPUTS: I : Ideal
EXAMPLE CODE:
```macaulay2
A = QQ<|a,b,c|>
B = QQ<|x,y|>
f = map(B,A,{x*y*x,y*x*y,x*y})
K = ncKernel f
``` | Key
ncKernel
(ncKernel,RingMap)
[ncKernel,DegreeLimit]
[ncKernel,Strategy]
Headline
Compute the graph ideal of a ring map between noncommutative rings.
Usage
I = ncKernel f
Inputs
f : RingMap
Outputs
I : Ideal
Description
Text
This function comp... | |||
doc_b5efc0b3_toRationalFunction | doc | toRationalFunction | Attempt to find a rational function representation. | output = toRationalFunction coeffs | A = QQ[x,y]/ideal{x^2,x*y}
kRes = res(coker vars A, LengthLimit => 10);
kBetti = apply(10, i -> numcols kRes.dd_i)
toRationalFunction kBetti | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Attempt to find a rational function representation.
USAGE: output = toRationalFunction coeffs
INPUTS: coeffs : List
OUTPUTS: output : Sequence
EXAMPLE CODE:
```macaulay2
A = QQ[x,y]/ideal{x^2,x*y}
kRes = res(coker vars A, LengthLimit => 10);
kBetti = apply(10, i -> numcols kRes.dd_i)
... | Key
toRationalFunction
(toRationalFunction, List)
Headline
Attempt to find a rational function representation.
Usage
output = toRationalFunction coeffs
Inputs
coeffs : List
Outputs
output : Sequence
Description
Text
This function attempts to find a rational funct... | |||
doc_b5efc0b3_(toRationalFunction,_List) | doc | (toRationalFunction, List) | Attempt to find a rational function representation. | output = toRationalFunction coeffs | A = QQ[x,y]/ideal{x^2,x*y}
kRes = res(coker vars A, LengthLimit => 10);
kBetti = apply(10, i -> numcols kRes.dd_i)
toRationalFunction kBetti | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Attempt to find a rational function representation.
USAGE: output = toRationalFunction coeffs
INPUTS: coeffs : List
OUTPUTS: output : Sequence
EXAMPLE CODE:
```macaulay2
A = QQ[x,y]/ideal{x^2,x*y}
kRes = res(coker vars A, LengthLimit => 10);
kBetti = apply(10, i -> numcols kRes.dd_i)
... | Key
toRationalFunction
(toRationalFunction, List)
Headline
Attempt to find a rational function representation.
Usage
output = toRationalFunction coeffs
Inputs
coeffs : List
Outputs
output : Sequence
Description
Text
This function attempts to find a rational funct... | |||
doc_e07f8771_pointScheme | doc | pointScheme | Compute the point scheme of the quadratic algebra B | I = pointScheme B | R = QQ[zz,X_1,X_2,X_3]
PP = sub(P,R) + ideal {zz^2 + zz + 1}
minPP = minimalPrimes PP; netList minPP
minPP / degree | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the point scheme of the quadratic algebra B
USAGE: I = pointScheme B
INPUTS: B : FreeAlgebraQuotient
OUTPUTS: I : Ideal
EXAMPLE CODE:
```macaulay2
R = QQ[zz,X_1,X_2,X_3]
PP = sub(P,R) + ideal {zz^2 + zz + 1}
minPP = minimalPrimes PP; netList minPP
minPP / degree
``` | Key
pointScheme
(pointScheme,FreeAlgebraQuotient,Symbol)
Headline
Compute the point scheme of the quadratic algebra B
Usage
I = pointScheme B
Inputs
B : FreeAlgebraQuotient
Outputs
I : Ideal
Description
Text
This method computes the ideal defining the point scheme ... | |||
doc_e07f8771_(pointScheme,FreeAlgebraQuotient,Symbol) | doc | (pointScheme,FreeAlgebraQuotient,Symbol) | Compute the point scheme of the quadratic algebra B | I = pointScheme B | R = QQ[zz,X_1,X_2,X_3]
PP = sub(P,R) + ideal {zz^2 + zz + 1}
minPP = minimalPrimes PP; netList minPP
minPP / degree | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the point scheme of the quadratic algebra B
USAGE: I = pointScheme B
INPUTS: B : FreeAlgebraQuotient
OUTPUTS: I : Ideal
EXAMPLE CODE:
```macaulay2
R = QQ[zz,X_1,X_2,X_3]
PP = sub(P,R) + ideal {zz^2 + zz + 1}
minPP = minimalPrimes PP; netList minPP
minPP / degree
``` | Key
pointScheme
(pointScheme,FreeAlgebraQuotient,Symbol)
Headline
Compute the point scheme of the quadratic algebra B
Usage
I = pointScheme B
Inputs
B : FreeAlgebraQuotient
Outputs
I : Ideal
Description
Text
This method computes the ideal defining the point scheme ... | |||
doc_f9f3a2dc_NCGB | doc | NCGB | Compute a two-sided Groebner basis of an ideal to a specified degree | Igb = NCGB(I,n) | A = ZZ/101<|x,y,z|>
I = ideal { x*y + y*x - 2*z^2,
y*z + z*y - 2*x^2,
z*x + x*z - 2*y^2}
Igb = NCGB(I,10) | "parallelism in engine computations" | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute a two-sided Groebner basis of an ideal to a specified degree
USAGE: Igb = NCGB(I,n)
INPUTS: I : Ideal
n : ZZ
Strategy => String
either "F4Parallel", "F4", or "Naive". Default for finite prime fields with
a homogeneous ideal is "F4Parallel", and otherwise default is "Naive", ... | Key
NCGB
(NCGB, Ideal)
(NCGB, Ideal, ZZ)
[NCGB,Strategy]
Headline
Compute a two-sided Groebner basis of an ideal to a specified degree
Usage
Igb = NCGB(I,n)
Inputs
I : Ideal
n : ZZ
Strategy => String
either "F4Parallel", "F4", or "Naive". Default for finite ... | ||
doc_f9f3a2dc_(NCGB,_Ideal) | doc | (NCGB, Ideal) | Compute a two-sided Groebner basis of an ideal to a specified degree | Igb = NCGB(I,n) | A = ZZ/101<|x,y,z|>
I = ideal { x*y + y*x - 2*z^2,
y*z + z*y - 2*x^2,
z*x + x*z - 2*y^2}
Igb = NCGB(I,10) | "parallelism in engine computations" | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute a two-sided Groebner basis of an ideal to a specified degree
USAGE: Igb = NCGB(I,n)
INPUTS: I : Ideal
n : ZZ
Strategy => String
either "F4Parallel", "F4", or "Naive". Default for finite prime fields with
a homogeneous ideal is "F4Parallel", and otherwise default is "Naive", ... | Key
NCGB
(NCGB, Ideal)
(NCGB, Ideal, ZZ)
[NCGB,Strategy]
Headline
Compute a two-sided Groebner basis of an ideal to a specified degree
Usage
Igb = NCGB(I,n)
Inputs
I : Ideal
n : ZZ
Strategy => String
either "F4Parallel", "F4", or "Naive". Default for finite ... | ||
doc_f9f3a2dc_(NCGB,_Ideal,_ZZ) | doc | (NCGB, Ideal, ZZ) | Compute a two-sided Groebner basis of an ideal to a specified degree | Igb = NCGB(I,n) | A = ZZ/101<|x,y,z|>
I = ideal { x*y + y*x - 2*z^2,
y*z + z*y - 2*x^2,
z*x + x*z - 2*y^2}
Igb = NCGB(I,10) | "parallelism in engine computations" | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute a two-sided Groebner basis of an ideal to a specified degree
USAGE: Igb = NCGB(I,n)
INPUTS: I : Ideal
n : ZZ
Strategy => String
either "F4Parallel", "F4", or "Naive". Default for finite prime fields with
a homogeneous ideal is "F4Parallel", and otherwise default is "Naive", ... | Key
NCGB
(NCGB, Ideal)
(NCGB, Ideal, ZZ)
[NCGB,Strategy]
Headline
Compute a two-sided Groebner basis of an ideal to a specified degree
Usage
Igb = NCGB(I,n)
Inputs
I : Ideal
n : ZZ
Strategy => String
either "F4Parallel", "F4", or "Naive". Default for finite ... | ||
doc_f9f3a2dc_[NCGB,Strategy] | doc | [NCGB,Strategy] | Compute a two-sided Groebner basis of an ideal to a specified degree | Igb = NCGB(I,n) | A = ZZ/101<|x,y,z|>
I = ideal { x*y + y*x - 2*z^2,
y*z + z*y - 2*x^2,
z*x + x*z - 2*y^2}
Igb = NCGB(I,10) | "parallelism in engine computations" | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute a two-sided Groebner basis of an ideal to a specified degree
USAGE: Igb = NCGB(I,n)
INPUTS: I : Ideal
n : ZZ
Strategy => String
either "F4Parallel", "F4", or "Naive". Default for finite prime fields with
a homogeneous ideal is "F4Parallel", and otherwise default is "Naive", ... | Key
NCGB
(NCGB, Ideal)
(NCGB, Ideal, ZZ)
[NCGB,Strategy]
Headline
Compute a two-sided Groebner basis of an ideal to a specified degree
Usage
Igb = NCGB(I,n)
Inputs
I : Ideal
n : ZZ
Strategy => String
either "F4Parallel", "F4", or "Naive". Default for finite ... | ||
doc_62bafe22_lineSchemeFourDim | doc | lineSchemeFourDim | Compute the line scheme of a four-dimensional AS regular algebra | L = lineSchemeFourDim B | R = QQ <|x_4,x_1,x_2,x_3|>
I = ideal {x_3^2 - x_1*x_2, x_4^2 - x_2*x_1, x_1*x_3 - x_2*x_4, x_3*x_1 - x_2*x_3, x_1*x_4 - x_4*x_2, x_4*x_1 - x_3*x_2}
Igb = NCGB(I, 10);
S = R/I
L = lineSchemeFourDim(S,M);
netList minimalPrimes L | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the line scheme of a four-dimensional AS regular algebra
USAGE: L = lineSchemeFourDim B
INPUTS: B : FreeAlgebraQuotient
OUTPUTS: L : Ideal
EXAMPLE CODE:
```macaulay2
R = QQ <|x_4,x_1,x_2,x_3|>
I = ideal {x_3^2 - x_1*x_2, x_4^2 - x_2*x_1, x_1*x_3 - x_2*x_4, x_3*x_1 - x_2*x_3, x_1*x_4 - x_4*x... | Key
lineSchemeFourDim
(lineSchemeFourDim,FreeAlgebraQuotient,Symbol)
Headline
Compute the line scheme of a four-dimensional AS regular algebra
Usage
L = lineSchemeFourDim B
Inputs
B : FreeAlgebraQuotient
Outputs
L : Ideal
Description
Text
This method computes the... | |||
doc_62bafe22_(lineSchemeFourDim,FreeAlgebraQuotient,Symbol) | doc | (lineSchemeFourDim,FreeAlgebraQuotient,Symbol) | Compute the line scheme of a four-dimensional AS regular algebra | L = lineSchemeFourDim B | R = QQ <|x_4,x_1,x_2,x_3|>
I = ideal {x_3^2 - x_1*x_2, x_4^2 - x_2*x_1, x_1*x_3 - x_2*x_4, x_3*x_1 - x_2*x_3, x_1*x_4 - x_4*x_2, x_4*x_1 - x_3*x_2}
Igb = NCGB(I, 10);
S = R/I
L = lineSchemeFourDim(S,M);
netList minimalPrimes L | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Compute the line scheme of a four-dimensional AS regular algebra
USAGE: L = lineSchemeFourDim B
INPUTS: B : FreeAlgebraQuotient
OUTPUTS: L : Ideal
EXAMPLE CODE:
```macaulay2
R = QQ <|x_4,x_1,x_2,x_3|>
I = ideal {x_3^2 - x_1*x_2, x_4^2 - x_2*x_1, x_1*x_3 - x_2*x_4, x_3*x_1 - x_2*x_3, x_1*x_4 - x_4*x... | Key
lineSchemeFourDim
(lineSchemeFourDim,FreeAlgebraQuotient,Symbol)
Headline
Compute the line scheme of a four-dimensional AS regular algebra
Usage
L = lineSchemeFourDim B
Inputs
B : FreeAlgebraQuotient
Outputs
L : Ideal
Description
Text
This method computes the... | |||
doc_9762d01f_freeProduct | doc | freeProduct | Define the free product of two algebras | C = freeProduct(A,B) | A = QQ<|x,y,z|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b,c})
C = freeProduct(A,B) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Define the free product of two algebras
USAGE: C = freeProduct(A,B)
INPUTS: A : Ring
B : Ring
OUTPUTS: C : FreeAlgebraQuotient
or @ TO FreeAlgebra @
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b,c})
C = freeProduct(A,B)
``` | Key
freeProduct
(freeProduct, Ring, Ring)
Headline
Define the free product of two algebras
Usage
C = freeProduct(A,B)
Inputs
A : Ring
B : Ring
Outputs
C : FreeAlgebraQuotient
or @ TO FreeAlgebra @
Description
Text
This function returns the free product of the alg... | |||
doc_9762d01f_(freeProduct,_Ring,_Ring) | doc | (freeProduct, Ring, Ring) | Define the free product of two algebras | C = freeProduct(A,B) | A = QQ<|x,y,z|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b,c})
C = freeProduct(A,B) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Define the free product of two algebras
USAGE: C = freeProduct(A,B)
INPUTS: A : Ring
B : Ring
OUTPUTS: C : FreeAlgebraQuotient
or @ TO FreeAlgebra @
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y,z|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b,c})
C = freeProduct(A,B)
``` | Key
freeProduct
(freeProduct, Ring, Ring)
Headline
Define the free product of two algebras
Usage
C = freeProduct(A,B)
Inputs
A : Ring
B : Ring
Outputs
C : FreeAlgebraQuotient
or @ TO FreeAlgebra @
Description
Text
This function returns the free product of the alg... | |||
doc_2e3324c0_qTensorProduct | doc | qTensorProduct | Define the (q-)commuting tensor product | C = qTensorProduct(A,B,q) | A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D = A ** B
ideal D | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Define the (q-)commuting tensor product
USAGE: C = qTensorProduct(A,B,q)
INPUTS: A : Ring
B : Ring
q : RingElement
OUTPUTS: C : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D... | Key
qTensorProduct
(qTensorProduct,Ring,Ring,ZZ)
(qTensorProduct,Ring,Ring,QQ)
(qTensorProduct,Ring,Ring,RingElement)
(symbol **, FreeAlgebra, FreeAlgebra)
(symbol **, FreeAlgebraQuotient, FreeAlgebra)
(symbol **, FreeAlgebra, FreeAlgebraQuotient)
(symbol **, FreeAlgebraQuotient, FreeAlg... | |||
doc_2e3324c0_(qTensorProduct,Ring,Ring,ZZ) | doc | (qTensorProduct,Ring,Ring,ZZ) | Define the (q-)commuting tensor product | C = qTensorProduct(A,B,q) | A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D = A ** B
ideal D | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Define the (q-)commuting tensor product
USAGE: C = qTensorProduct(A,B,q)
INPUTS: A : Ring
B : Ring
q : RingElement
OUTPUTS: C : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D... | Key
qTensorProduct
(qTensorProduct,Ring,Ring,ZZ)
(qTensorProduct,Ring,Ring,QQ)
(qTensorProduct,Ring,Ring,RingElement)
(symbol **, FreeAlgebra, FreeAlgebra)
(symbol **, FreeAlgebraQuotient, FreeAlgebra)
(symbol **, FreeAlgebra, FreeAlgebraQuotient)
(symbol **, FreeAlgebraQuotient, FreeAlg... | |||
doc_2e3324c0_(qTensorProduct,Ring,Ring,QQ) | doc | (qTensorProduct,Ring,Ring,QQ) | Define the (q-)commuting tensor product | C = qTensorProduct(A,B,q) | A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D = A ** B
ideal D | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Define the (q-)commuting tensor product
USAGE: C = qTensorProduct(A,B,q)
INPUTS: A : Ring
B : Ring
q : RingElement
OUTPUTS: C : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D... | Key
qTensorProduct
(qTensorProduct,Ring,Ring,ZZ)
(qTensorProduct,Ring,Ring,QQ)
(qTensorProduct,Ring,Ring,RingElement)
(symbol **, FreeAlgebra, FreeAlgebra)
(symbol **, FreeAlgebraQuotient, FreeAlgebra)
(symbol **, FreeAlgebra, FreeAlgebraQuotient)
(symbol **, FreeAlgebraQuotient, FreeAlg... | |||
doc_2e3324c0_(qTensorProduct,Ring,Ring,RingElement) | doc | (qTensorProduct,Ring,Ring,RingElement) | Define the (q-)commuting tensor product | C = qTensorProduct(A,B,q) | A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D = A ** B
ideal D | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Define the (q-)commuting tensor product
USAGE: C = qTensorProduct(A,B,q)
INPUTS: A : Ring
B : Ring
q : RingElement
OUTPUTS: C : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D... | Key
qTensorProduct
(qTensorProduct,Ring,Ring,ZZ)
(qTensorProduct,Ring,Ring,QQ)
(qTensorProduct,Ring,Ring,RingElement)
(symbol **, FreeAlgebra, FreeAlgebra)
(symbol **, FreeAlgebraQuotient, FreeAlgebra)
(symbol **, FreeAlgebra, FreeAlgebraQuotient)
(symbol **, FreeAlgebraQuotient, FreeAlg... | |||
doc_2e3324c0_(symbol_**,_FreeAlgebra,_FreeAlgebra) | doc | (symbol **, FreeAlgebra, FreeAlgebra) | Define the (q-)commuting tensor product | C = qTensorProduct(A,B,q) | A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D = A ** B
ideal D | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Define the (q-)commuting tensor product
USAGE: C = qTensorProduct(A,B,q)
INPUTS: A : Ring
B : Ring
q : RingElement
OUTPUTS: C : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D... | Key
qTensorProduct
(qTensorProduct,Ring,Ring,ZZ)
(qTensorProduct,Ring,Ring,QQ)
(qTensorProduct,Ring,Ring,RingElement)
(symbol **, FreeAlgebra, FreeAlgebra)
(symbol **, FreeAlgebraQuotient, FreeAlgebra)
(symbol **, FreeAlgebra, FreeAlgebraQuotient)
(symbol **, FreeAlgebraQuotient, FreeAlg... | |||
doc_2e3324c0_(symbol_**,_FreeAlgebraQuotient,_FreeAlgebra) | doc | (symbol **, FreeAlgebraQuotient, FreeAlgebra) | Define the (q-)commuting tensor product | C = qTensorProduct(A,B,q) | A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D = A ** B
ideal D | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Define the (q-)commuting tensor product
USAGE: C = qTensorProduct(A,B,q)
INPUTS: A : Ring
B : Ring
q : RingElement
OUTPUTS: C : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D... | Key
qTensorProduct
(qTensorProduct,Ring,Ring,ZZ)
(qTensorProduct,Ring,Ring,QQ)
(qTensorProduct,Ring,Ring,RingElement)
(symbol **, FreeAlgebra, FreeAlgebra)
(symbol **, FreeAlgebraQuotient, FreeAlgebra)
(symbol **, FreeAlgebra, FreeAlgebraQuotient)
(symbol **, FreeAlgebraQuotient, FreeAlg... | |||
doc_2e3324c0_(symbol_**,_FreeAlgebra,_FreeAlgebraQuotient) | doc | (symbol **, FreeAlgebra, FreeAlgebraQuotient) | Define the (q-)commuting tensor product | C = qTensorProduct(A,B,q) | A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D = A ** B
ideal D | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Define the (q-)commuting tensor product
USAGE: C = qTensorProduct(A,B,q)
INPUTS: A : Ring
B : Ring
q : RingElement
OUTPUTS: C : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D... | Key
qTensorProduct
(qTensorProduct,Ring,Ring,ZZ)
(qTensorProduct,Ring,Ring,QQ)
(qTensorProduct,Ring,Ring,RingElement)
(symbol **, FreeAlgebra, FreeAlgebra)
(symbol **, FreeAlgebraQuotient, FreeAlgebra)
(symbol **, FreeAlgebra, FreeAlgebraQuotient)
(symbol **, FreeAlgebraQuotient, FreeAlg... | |||
doc_2e3324c0_(symbol_**,_FreeAlgebraQuotient,_FreeAlgebraQuotient) | doc | (symbol **, FreeAlgebraQuotient, FreeAlgebraQuotient) | Define the (q-)commuting tensor product | C = qTensorProduct(A,B,q) | A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D = A ** B
ideal D | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Define the (q-)commuting tensor product
USAGE: C = qTensorProduct(A,B,q)
INPUTS: A : Ring
B : Ring
q : RingElement
OUTPUTS: C : FreeAlgebraQuotient
EXAMPLE CODE:
```macaulay2
A = QQ<|x,y|>
B = skewPolynomialRing(QQ,(-1)_QQ, {a,b})
C = qTensorProduct(A,B,-1_QQ)
ideal C
D... | Key
qTensorProduct
(qTensorProduct,Ring,Ring,ZZ)
(qTensorProduct,Ring,Ring,QQ)
(qTensorProduct,Ring,Ring,RingElement)
(symbol **, FreeAlgebra, FreeAlgebra)
(symbol **, FreeAlgebraQuotient, FreeAlgebra)
(symbol **, FreeAlgebra, FreeAlgebraQuotient)
(symbol **, FreeAlgebraQuotient, FreeAlg... | |||
doc_48282a9c_rightKernel | doc | rightKernel | Right kernel of a matrix | K = rightKernel M | d1*d2
ncMatrixMult(d1,d2) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Right kernel of a matrix
USAGE: K = rightKernel M
INPUTS: M : Matrix
OUTPUTS: N : Matrix
EXAMPLE CODE:
```macaulay2
d1*d2
ncMatrixMult(d1,d2)
``` | Key
rightKernel
(rightKernel,Matrix)
[rightKernel,DegreeLimit]
[rightKernel,Strategy]
Headline
Right kernel of a matrix
Usage
K = rightKernel M
Inputs
M : Matrix
Outputs
N : Matrix
Description
Text
This function computes a minimal generating set of th... | |||
doc_48282a9c_(rightKernel,Matrix) | doc | (rightKernel,Matrix) | Right kernel of a matrix | K = rightKernel M | d1*d2
ncMatrixMult(d1,d2) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Right kernel of a matrix
USAGE: K = rightKernel M
INPUTS: M : Matrix
OUTPUTS: N : Matrix
EXAMPLE CODE:
```macaulay2
d1*d2
ncMatrixMult(d1,d2)
``` | Key
rightKernel
(rightKernel,Matrix)
[rightKernel,DegreeLimit]
[rightKernel,Strategy]
Headline
Right kernel of a matrix
Usage
K = rightKernel M
Inputs
M : Matrix
Outputs
N : Matrix
Description
Text
This function computes a minimal generating set of th... | |||
doc_48282a9c_[rightKernel,DegreeLimit] | doc | [rightKernel,DegreeLimit] | Right kernel of a matrix | K = rightKernel M | d1*d2
ncMatrixMult(d1,d2) | M2_git/M2/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2 | stable | HEADLINE: Right kernel of a matrix
USAGE: K = rightKernel M
INPUTS: M : Matrix
OUTPUTS: N : Matrix
EXAMPLE CODE:
```macaulay2
d1*d2
ncMatrixMult(d1,d2)
``` | Key
rightKernel
(rightKernel,Matrix)
[rightKernel,DegreeLimit]
[rightKernel,Strategy]
Headline
Right kernel of a matrix
Usage
K = rightKernel M
Inputs
M : Matrix
Outputs
N : Matrix
Description
Text
This function computes a minimal generating set of th... |
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