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In mathematics, especially group theory, the centralizer (also called commutant) of a subset S in a group G is the set
C
G
(
S
)
{\displaystyle \operatorname {C} _{G}(S)}
of elements of G that commute with every element of S, or equivalently, such that conjugation by
g
{\displaystyle g}
leaves each element of S fixed. The normalizer of S in G is the set of elements
N
G
(
S
)
{\displaystyle \mathrm {N} _{G}(S)}
of G that satisfy the weaker condition of leaving the set
S
⊆
G
{\displaystyle S\subseteq G}
fixed under conjugation. The centralizer and normalizer of S are subgroups of G | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero.
That is, char(R) is the smallest positive number n such that:(p 198, Thm | https://huggingface.co/datasets/fmars/wiki_stem |
In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified. In each case, the Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two copies of such an algebra, though not in a canonical way. Below it is shown that distinct Clifford algebras may be algebra-isomorphic, as is the case of Cl1,1(R) and Cl2,0(R), which are both isomorphic as rings to the ring of two-by-two matrices over the real numbers | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a clean ring is a ring in which every element can be written as the sum of a unit and an idempotent. A ring is a local ring if and only if it is clean and has no idempotents other than 0 and 1. The endomorphism ring of a continuous module is a clean ring | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a (left) coherent ring is a ring in which every finitely generated left ideal is finitely presented.
Many theorems about finitely generated modules over Noetherian rings can be extended to finitely presented modules over coherent rings.
Every left Noetherian ring is left coherent | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings | https://huggingface.co/datasets/fmars/wiki_stem |
In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary (proposed by Donald Knuth in 1955) or complex number (proposed by S. Khmelnik in 1964 and Walter F. Penney in 1965) | https://huggingface.co/datasets/fmars/wiki_stem |
In ring theory, a branch of mathematics, the conductor is a measurement of how far apart a commutative ring and an extension ring are. Most often, the larger ring is a domain integrally closed in its field of fractions, and then the conductor measures the failure of the smaller ring to be integrally closed.
The conductor is of great importance in the study of non-maximal orders in the ring of integers of an algebraic number field | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, especially in the field of commutative algebra, a connected ring is a commutative ring A that satisfies one of the following equivalent conditions:
A possesses no non-trivial (that is, not equal to 1 or 0) idempotent elements;
the spectrum of A with the Zariski topology is a connected space.
Examples and non-examples
Connectedness defines a fairly general class of commutative rings. For example, all local rings and all (meet-)irreducible rings are connected | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a ring is said to be a Dedekind-finite ring if ab = 1 implies ba = 1 for any two ring elements a and b. In other words, all one-sided inverses in the ring are two-sided.
These rings have also been called directly finite rings and von Neumann finite rings | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, in particular the study of abstract algebra, a Dedekind–Hasse norm is a function on an integral domain that generalises the notion of a Euclidean function on Euclidean domains.
Definition
Let R be an integral domain and g : R → Z≥0 be a function from R to the non-negative integers. Denote by 0R the additive identity of R | https://huggingface.co/datasets/fmars/wiki_stem |
In abstract algebra, specifically in module theory, a dense submodule of a module is a refinement of the notion of an essential submodule. If N is a dense submodule of M, it may alternatively be said that "N ⊆ M is a rational extension". Dense submodules are connected with rings of quotients in noncommutative ring theory | https://huggingface.co/datasets/fmars/wiki_stem |
In ring theory and Frobenius algebra extensions, areas of mathematics, there is a notion of depth two subring or depth of a Frobenius extension. The notion of depth two is important in a certain noncommutative Galois theory, which generates Hopf algebroids in place of the more classical Galois groups, whereas the notion of depth greater than two measures the defect, or distance, from being depth two in a tower of iterated endomorphism rings above the subring. A more recent definition of depth of any unital subring in any associative ring is proposed (see below) in a paper studying the depth of a subgroup of a finite group as group algebras over a commutative ring | https://huggingface.co/datasets/fmars/wiki_stem |
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over
Q
{\displaystyle \mathbb {Q} }
), simplicial commutative rings or
E
∞
{\displaystyle E_{\infty }}
-ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e. g. , Tor) of the structure sheaf | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. With the development of abstract rings, of which the integers are the archetype, the original notion of divisor found a natural extension.
Divisibility is a useful concept for the analysis of the structure of commutative rings because of its relationship with the ideal structure of such rings | https://huggingface.co/datasets/fmars/wiki_stem |
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra D over a field. We call D a division algebra if for any element a in D and any non-zero element b in D there exists precisely one element x in D with a = bx and precisely one element y in D such that a = yb | https://huggingface.co/datasets/fmars/wiki_stem |
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element a has a multiplicative inverse, that is, an element usually denoted a–1, such that a a–1 = a–1 a = 1. So, (right) division may be defined as a / b = a b–1, but this notation is avoided, as one may have a b–1 ≠ b–1 a | https://huggingface.co/datasets/fmars/wiki_stem |
In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. (Sometimes such a ring is said to "have the zero-product property". ) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor) | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in computer science because they are the only ones with finite binary representations | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination
of them (Bézout's identity) | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements.
Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite group, but the concept of finite rings in their own right has a more recent history.
Although rings have more structure than groups, the theory of finite rings is simpler than that of finite groups | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc. ).
A formal power series is a special kind of formal series, whose terms are of the form
a
x
n
{\displaystyle ax^{n}}
where
x
n
{\displaystyle x^{n}}
is the
n
{\displaystyle n}
th power of a variable
x
{\displaystyle x}
(
n
{\displaystyle n}
is a non-negative integer), and
a
{\displaystyle a}
is called the coefficient | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded as a free commutative algebra.
Definition
For R a commutative ring, the free (associative, unital) algebra on n indeterminates {X1, | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, especially in the field of ring theory, a (right) free ideal ring, or fir, is a ring in which all right ideals are free modules with unique rank. A ring such that all right ideals with at most n generators are free and have unique rank is called an n-fir. A semifir is a ring in which all finitely generated right ideals are free modules of unique rank | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i. e. , there is a unique minimal principal ideal containing the ideal generated by two given elements | https://huggingface.co/datasets/fmars/wiki_stem |
In ring theory and homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of a ring A denoted gl dim A, is a non-negative integer or infinity which is a homological invariant of the ring. It is defined to be the supremum of the set of projective dimensions of all A-modules. Global dimension is an important technical notion in the dimension theory of Noetherian rings | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a Goldman domain or G-domain is an integral domain A whose field of fractions is a finitely generated algebra over A. They are named after Oscar Goldman.
An overring (i | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups
R
i
{\displaystyle R_{i}}
such that
R
i
R
j
⊆
R
i
+
j
{\displaystyle R_{i}R_{j}\subseteq R_{i+j}}
. The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading | https://huggingface.co/datasets/fmars/wiki_stem |
In algebra, given a commutative ring R, the graded-symmetric algebra of a graded R-module M is the quotient of the tensor algebra of M by the ideal I generated by elements of the form:
x
y
−
(
−
1
)
|
x
|
|
y
|
y
x
{\displaystyle xy-(-1)^{|x||y|}yx}
x
2
{\displaystyle x^{2}}
when |x | is oddfor homogeneous elements x, y in M of degree |x |, |y |. By construction, a graded-symmetric algebra is graded-commutative; i. e | https://huggingface.co/datasets/fmars/wiki_stem |
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, especially in the area of abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective. If this is required only for finitely generated submodules, it is called semihereditary.
For a noncommutative ring R, the terms left hereditary and left semihereditary and their right hand versions are used to distinguish the property on a single side of the ring | https://huggingface.co/datasets/fmars/wiki_stem |
In algebra, the term Hermite ring (after Charles Hermite) has been applied to three different objects.
According to Kaplansky (1949) (p. 465), a ring is right Hermite if, for every two elements a and b of the ring, there is an element d of the ring and an invertible 2 by 2 matrix M over the ring such that (a b)M=(d 0) | https://huggingface.co/datasets/fmars/wiki_stem |
In algebra, the Hilbert–Kunz function of a local ring (R, m) of prime characteristic p is the function
f
(
q
)
=
length
R
(
R
/
m
[
q
]
)
{\displaystyle f(q)=\operatorname {length} _{R}(R/m^{[q]})}
where q is a power of p and m[q] is the ideal generated by the q-th powers of elements of the maximal ideal m. The notion was introduced by Ernst Kunz, who used it to characterize a regular ring as a Noetherian ring in which the Frobenius morphism is flat. If d is the dimension of the local ring, Monsky showed that f(q)/(q^d) is c+O(1/q) for some real constant c | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by Gerhard Hochschild (1945) for algebras over a field, and extended to algebras over more general rings by Henri Cartan and Samuel Eilenberg (1956) | https://huggingface.co/datasets/fmars/wiki_stem |
In the branch of mathematics called category theory, a hopfian object is an object A such that any epimorphism of A onto A is necessarily an automorphism. The dual notion is that of a cohopfian object, which is an object B such that every monomorphism from B into B is necessarily an automorphism. The two conditions have been studied in the categories of groups, rings, modules, and topological spaces | https://huggingface.co/datasets/fmars/wiki_stem |
In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that a = a2 = a3 = a4 = | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, an integer-valued polynomial (also known as a numerical polynomial)
P
(
t
)
{\displaystyle P(t)}
is a polynomial whose value
P
(
n
)
{\displaystyle P(n)}
is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true. For example, the polynomial
1
2
t
2
+
1
2
t
=
1
2
t
(
t
+
1
)
{\displaystyle {\frac {1}{2}}t^{2}+{\frac {1}{2}}t={\frac {1}{2}}t(t+1)}
takes on integer values whenever t is an integer | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c | https://huggingface.co/datasets/fmars/wiki_stem |
In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and aj in A such that
b
n
+
a
n
−
1
b
n
−
1
+
⋯
+
a
1
b
+
a
0
=
0.
{\displaystyle b^{n}+a_{n-1}b^{n-1}+\cdots +a_{1}b+a_{0}=0. }
That is to say, b is a root of a monic polynomial over A | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, more specifically in the field of ring theory, a ring has the invariant basis number (IBN) property if all finitely generated free left modules over R have a well-defined rank. In the case of fields, the IBN property becomes the statement that finite-dimensional vector spaces have a unique dimension.
Definition
A ring R has invariant basis number (IBN) if for all positive integers m and n, Rm isomorphic to Rn (as left R-modules) implies that m = n | https://huggingface.co/datasets/fmars/wiki_stem |
In algebra, an irreducible element of an integral domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements.
The irreducible elements are the terminal elements of a factorization process; that is, they are the factors that cannot be further factorized. The irreducible factors of an element are uniquely defined, up to the multiplication by a unit, if the integral domain is a unique factorization domain | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a proper ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two strictly larger ideals.
Examples
Every prime ideal is irreducible. Let
J
{\displaystyle J}
and
K
{\displaystyle K}
be ideals of a commutative ring
R
{\displaystyle R}
, with neither one contained in the other | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, especially in the field of ring theory, the term irreducible ring is used in a few different ways.
A (meet-)irreducible ring is a ring in which the intersection of two non-zero ideals is always non-zero.
A directly irreducible ring is a ring which cannot be written as the direct sum of two non-zero rings | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions. They are named for Paul Jaffard who first studied them in 1960.
Formally, a Jaffard ring is a ring R such that the polynomial ring
dim
R
[
T
1
,
…
,
T
n
]
=
n
+
dim
R
,
{\displaystyle \dim R[T_{1},\ldots ,T_{n}]=n+\dim R,\,}
where "dim" denotes Krull dimension | https://huggingface.co/datasets/fmars/wiki_stem |
The mathematician Irving Kaplansky is notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras. They are usually known as Kaplansky's conjectures.
Group rings
Let K be a field, and G a torsion-free group | https://huggingface.co/datasets/fmars/wiki_stem |
In ring theory, a subfield of abstract algebra, a right Kasch ring is a ring R for which every simple right R module is isomorphic to a right ideal of R. Analogously the notion of a left Kasch ring is defined, and the two properties are independent of each other.
Kasch rings are named in honor of mathematician Friedrich Kasch | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematical cryptography, a Kleinian integer is a complex number of the form
m
+
n
1
+
−
7
2
{\displaystyle m+n{\frac {1+{\sqrt {-7}}}{2}}}
, with m and n rational integers. They are named after Felix Klein.
The Kleinian integers form a ring called the Kleinian ring, which is the ring of integers in the imaginary quadratic field
Q
(
−
7
)
{\displaystyle \mathbb {Q} ({\sqrt {-7}})}
| https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, the Köthe conjecture is a problem in ring theory, open as of 2022. It is formulated in various ways. Suppose that R is a ring | https://huggingface.co/datasets/fmars/wiki_stem |
In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λn on it that behave like the exterior powers of vector spaces. Many rings considered in K-theory carry a natural λ-ring structure. λ-rings also provide a powerful formalism for studying an action of the symmetric functions on the ring of polynomials, recovering and extending many classical results (Lascoux (2003)) | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a Laurent polynomial (named
after Pierre Alphonse Laurent) in one variable over a field
F
{\displaystyle \mathbb {F} }
is a linear combination of positive and negative powers of the variable with coefficients in
F
{\displaystyle \mathbb {F} }
. Laurent polynomials in X form a ring denoted
F
[
X
,
X
−
1
]
{\displaystyle \mathbb {F} [X,X^{-1}]}
. They differ from ordinary polynomials in that they may have terms of negative degree | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules.
In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal | https://huggingface.co/datasets/fmars/wiki_stem |
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of fractions
m
s
,
{\displaystyle {\frac {m}{s}},}
such that the denominator s belongs to a given subset S of R. If S is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field
Q
{\displaystyle \mathbb {Q} }
of rational numbers from the ring
Z
{\displaystyle \mathbb {Z} }
of integers | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a Loewy ring or semi-Artinian ring is a ring in which every non-zero module has a non-zero socle, or equivalently if the Loewy length of every module is defined. The concepts are named after Alfred Loewy.
Loewy length
The Loewy length and Loewy series were introduced by Emil Artin, Cecil J | https://huggingface.co/datasets/fmars/wiki_stem |
In abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication (Lam 1999). The set of all n × n matrices with entries in R is a matrix ring denoted Mn(R) (alternative notations: Matn(R) and Rn×n). Some sets of infinite matrices form infinite matrix rings | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R.
Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields | https://huggingface.co/datasets/fmars/wiki_stem |
In the branch of abstract algebra known as ring theory, a minimal right ideal of a ring R is a non-zero right ideal which contains no other non-zero right ideal. Likewise, a minimal left ideal is a non-zero left ideal of R containing no other non-zero left ideals of R, and a minimal ideal of R is a non-zero ideal containing no other non-zero two-sided ideal of R (Isaacs 2009, p. 190) | https://huggingface.co/datasets/fmars/wiki_stem |
In abstract algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group.
Definition
Let R be a ring and let G be a monoid. The monoid ring or monoid algebra of G over R, denoted R[G] or RG, is the set of formal sums
∑
g
∈
G
r
g
g
{\displaystyle \sum _{g\in G}r_{g}g}
,
where
r
g
∈
R
{\displaystyle r_{g}\in R}
for each
g
∈
G
{\displaystyle g\in G}
and rg = 0 for all but finitely many g, equipped with coefficient-wise addition, and the multiplication in which the elements of R commute with the elements of G | https://huggingface.co/datasets/fmars/wiki_stem |
In algebra, a Nakayama algebra or generalized uniserial algebra is an algebra such that each left or right indecomposable projective module has a unique composition series. They were studied by Tadasi Nakayama (1940) who called them "generalized uni-serial rings". These algebras were further studied by Herbert Kupisch (1959) and later by Ichiro Murase (1963-64), by Kent Ralph Fuller (1968) and by Idun Reiten (1982) | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, Nakayama's conjecture is a conjecture about Artinian rings, introduced by Nakayama (1958). The generalized Nakayama conjecture is an extension to more general rings, introduced by Auslander and Reiten (1975). Leuschke & Huneke (2004) proved some cases of the generalized Nakayama conjecture | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups.
Definition
A set N together with two binary operations + (called addition) and ⋅ (called multiplication) is called a (right) near-ring if:
N is a group (not necessarily abelian) under addition;
multiplication is associative (so N is a semigroup under multiplication); and
multiplication on the right distributes over addition: for any x, y, z in N, it holds that (x + y)⋅z = (x⋅z) + (y⋅z) | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, the necklace ring is a ring introduced by Metropolis and Rota (1983) to elucidate the multiplicative properties of necklace polynomials.
Definition
If A is a commutative ring then the necklace ring over A consists of all infinite sequences
(
a
1
,
a
2
,
.
| https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer n every product containing at least n elements of the algebra is zero. The concept of a nilpotent Lie algebra has a different definition, which depends upon the Lie bracket. (There is no Lie bracket for many algebras over commutative rings; a Lie algebra involves its Lie bracket, whereas, there is no Lie bracket defined in the general case of an algebra over a commutative ring | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence
I
1
⊆
I
2
⊆
I
3
⊆
⋯
{\displaystyle I_{1}\subseteq I_{2}\subseteq I_{3}\subseteq \cdots }
of left (or right) ideals has a largest element; that is, there exists an n such that:
I
n
=
I
n
+
1
=
⋯
.
{\displaystyle I_{n}=I_{n+1}=\cdots | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring is a ring that is not a commutative ring.
Noncommutative algebra is the part of ring theory devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings | https://huggingface.co/datasets/fmars/wiki_stem |
A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix β > 1, the value of
x
=
d
n
…
d
2
d
1
d
0
.
d
−
1
d
−
2
…
d
−
m
{\displaystyle x=d_{n}\dots d_{2}d_{1}d_{0} | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a noncommutative unique factorization domain is a noncommutative ring with the unique factorization property.
Examples
The ring of Hurwitz quaternions, also known as integral quaternions. A quaternion a = a0 + a1i + a2j + a3k is integral if either all the coefficients ai are integers or all of them are half-integers | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring (R, +, ⋅) is the ring (R, +, ∗) whose multiplication ∗ is defined by a ∗ b = b ⋅ a for all a, b in R. The opposite ring can be used to define multimodules, a generalization of bimodules | https://huggingface.co/datasets/fmars/wiki_stem |
In computer algebra, an Ore algebra is a special kind of iterated Ore extension that can be used to represent linear functional operators, including linear differential and/or recurrence operators. The concept is named after Øystein Ore.
Definition
Let
K
{\displaystyle K}
be a (commutative) field and
A
=
K
[
x
1
,
…
,
x
s
]
{\displaystyle A=K[x_{1},\ldots ,x_{s}]}
be a commutative polynomial ring (with
A
=
K
{\displaystyle A=K}
when
s
=
0
{\displaystyle s=0}
) | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The right Ore condition for a multiplicative subset S of a ring R is that for a ∈ R and s ∈ S, the intersection aS ∩ sR ≠ ∅. A (non-commutative) domain for which the set of non-zero elements satisfies the right Ore condition is called a right Ore domain | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Øystein Ore, is a special type of a ring extension whose properties are relatively well understood. Elements of a Ore extension are called Ore polynomials.
Ore extensions appear in several natural contexts, including skew and differential polynomial rings, group algebras of polycyclic groups, universal enveloping algebras of solvable Lie algebras, and coordinate rings of quantum groups | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, an overring of an integral domain contains the integral domain, and the integral domain's field of fractions contains the overring. Overrings provide an improved understanding of different types of rings and domains.
Definition
In this article, all rings are commutative rings, and ring and overring share the same identity element | https://huggingface.co/datasets/fmars/wiki_stem |
In abstract algebra, a partially ordered ring is a ring (A, +, ·), together with a compatible partial order, that is, a partial order
≤
{\displaystyle \,\leq \,}
on the underlying set A that is compatible with the ring operations in the sense that it satisfies:
and
for all
x
,
y
,
z
∈
A
{\displaystyle x,y,z\in A}
. Various extensions of this definition exist that constrain the ring, the partial order, or both. For example, an Archimedean partially ordered ring is a partially ordered ring
(
A
,
≤
)
{\displaystyle (A,\leq )}
where
A
{\displaystyle A}
's partially ordered additive group is Archimedean | https://huggingface.co/datasets/fmars/wiki_stem |
In algebra, a field k is perfect if any one of the following equivalent conditions holds:
Every irreducible polynomial over k has distinct roots.
Every irreducible polynomial over k is separable.
Every finite extension of k is separable | https://huggingface.co/datasets/fmars/wiki_stem |
In the area of abstract algebra known as ring theory, a left perfect ring is a type of ring in which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, there exist rings which are perfect on one side but not the other. Perfect rings were introduced in Bass's book | https://huggingface.co/datasets/fmars/wiki_stem |
In mathematics, a Poisson ring is a commutative ring on which an anticommutative and distributive binary operation
[
⋅
,
⋅
]
{\displaystyle [\cdot ,\cdot ]}
satisfying the Jacobi identity and the product rule is defined. Such an operation is then known as the Poisson bracket of the Poisson ring.
Many important operations and results of symplectic geometry and Hamiltonian mechanics may be formulated in terms of the Poisson bracket and, hence, apply to Poisson algebras as well | https://huggingface.co/datasets/fmars/wiki_stem |
In algebra, the ring of polynomial differential forms on the standard n-simplex is the differential graded algebra:
Ω
poly
∗
(
[
n
]
)
=
Q
[
t
0
,
.
.
| https://huggingface.co/datasets/fmars/wiki_stem |
In the physics, the Matthias rules refers to a historical set of empirical guidelines on how to find superconductors. These rules were authored Bernd T. Matthias who discovered hundreds of superconductors using these principles in the 1950s and 1960s | https://huggingface.co/datasets/fmars/wiki_stem |
In the beginning of the 19th century, many experimental and theoretical works had been accomplished in the understanding of electromagnetics. In the 1780s, Coulomb's law of electrostatics had been established. In 1825, Ampère published his Ampère's law | https://huggingface.co/datasets/fmars/wiki_stem |
In the history of science, the mechanical equivalent of heat states that motion and heat are mutually interchangeable and that in every case, a given amount of work would generate the same amount of heat, provided the work done is totally converted to heat energy. The mechanical equivalent of heat was a concept that had an important part in the development and acceptance of the conservation of energy and the establishment of the science of thermodynamics in the 19th century.
History and priority dispute
Benjamin Thompson, Count Rumford, had observed the frictional heat generated by boring cannon at the arsenal in Munich, Bavaria, circa 1797 Rumford immersed a cannon barrel in water and arranged for a specially blunted boring tool | https://huggingface.co/datasets/fmars/wiki_stem |
Mechanical explanations of gravitation (or kinetic theories of gravitation) are attempts to explain the action of gravity by aid of basic mechanical processes, such as pressure forces caused by pushes, without the use of any action at a distance. These theories were developed from the 16th until the 19th century in connection with the aether. However, such models are no longer regarded as viable theories within the mainstream scientific community and general relativity is now the standard model to describe gravitation without the use of actions at a distance | https://huggingface.co/datasets/fmars/wiki_stem |
Moseley's law is an empirical law concerning the characteristic X-rays emitted by atoms. The law had been discovered and published by the English physicist Henry Moseley in 1913–1914. Until Moseley's work, "atomic number" was merely an element's place in the periodic table and was not known to be associated with any measurable physical quantity | https://huggingface.co/datasets/fmars/wiki_stem |
In the study of heat transfer, Newton's law of cooling is a physical law which states that
The rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its environment.
The law is frequently qualified to include the condition that the temperature difference is small and the nature of heat transfer mechanism remains the same. As such, it is equivalent to a statement that the heat transfer coefficient, which mediates between heat losses and temperature differences, is a constant | https://huggingface.co/datasets/fmars/wiki_stem |
The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory is now understood as the semi-classical approximation to modern quantum mechanics | https://huggingface.co/datasets/fmars/wiki_stem |
The Oxford Calculators were a group of 14th-century thinkers, almost all associated with Merton College, Oxford; for this reason they were dubbed "The Merton School". These men took a strikingly logical and mathematical approach to philosophical problems.
The key "calculators", writing in the second quarter of the 14th century, were Thomas Bradwardine, William Heytesbury, Richard Swineshead and John Dumbleton | https://huggingface.co/datasets/fmars/wiki_stem |
The Oxford Electric Bell or Clarendon Dry Pile is an experimental electric bell, in particular a type of bell that uses the electrostatic clock principle that was set up in 1840 and which has run nearly continuously ever since. It was one of the first pieces purchased for a collection of apparatus by clergyman and physicist Robert Walker. It is located in a corridor adjacent to the foyer of the Clarendon Laboratory at the University of Oxford, England, and is still ringing, albeit inaudibly due to being behind two layers of glass | https://huggingface.co/datasets/fmars/wiki_stem |
The periodic table is an arrangement of the chemical elements, structured by their atomic number, electron configuration and recurring chemical properties. In the basic form, elements are presented in order of increasing atomic number, in the reading sequence. Then, rows and columns are created by starting new rows and inserting blank cells, so that rows (periods) and columns (groups) show elements with recurring properties (called periodicity) | https://huggingface.co/datasets/fmars/wiki_stem |
The natural sciences saw various advancements during the Golden Age of Islam (from roughly the mid 8th to the mid 13th centuries), adding a number of innovations to the Transmission of the Classics (such as Aristotle, Ptolemy, Euclid, Neoplatonism). During this period, Islamic theology was encouraging of thinkers to find knowledge. Thinkers from this period included Al-Farabi, Abu Bishr Matta, Ibn Sina, al-Hassan Ibn al-Haytham and Ibn Bajjah | https://huggingface.co/datasets/fmars/wiki_stem |
Physics outreach encompasses facets of science outreach and physics education, and a variety of activities by schools, research institutes, universities, clubs and institutions such as science museums aimed at broadening the audience for and awareness and understanding of physics. While the general public may sometimes be the focus of such activities, physics outreach often centers on developing and providing resources and making presentations to students, educators in other disciplines, and in some cases researchers within different areas of physics.
History
Ongoing efforts to expand the understanding of physics to a wider audience have been undertaken by individuals and institutions since the early 19th century | https://huggingface.co/datasets/fmars/wiki_stem |
For light and other electromagnetic radiation, the plane of polarization is the plane spanned by the direction of propagation and either the electric vector or the magnetic vector, depending on the convention. It can be defined for polarized light, remains fixed in space for linearly-polarized light, and undergoes axial rotation for circularly-polarized light.
Unfortunately the two conventions are contradictory | https://huggingface.co/datasets/fmars/wiki_stem |
The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the action of a mechanical system, yields the equations of motion for that system. The principle states that the trajectories (i. e | https://huggingface.co/datasets/fmars/wiki_stem |
Quaestiones quaedam philosophicae (Certain philosophical questions) is the name given to a set of notes that Isaac Newton kept for himself during his earlier years in Cambridge. They concern questions in the natural philosophy of the day that interested him. Apart from the light it throws on the formation of his own agenda for research, the major interest in these notes is the documentation of the unaided development of the scientific method in the mind of Newton, whereby every question is put to experimental test | https://huggingface.co/datasets/fmars/wiki_stem |
In particle physics, the history of quantum field theory starts with its creation by Paul Dirac, when he attempted to quantize the electromagnetic field in the late 1920s. Heisenberg was awarded the 1932 Nobel Prize in Physics "for the creation of quantum mechanics". Major advances in the theory were made in the 1940s and 1950s, leading to the introduction of renormalized quantum electrodynamics (QED) | https://huggingface.co/datasets/fmars/wiki_stem |
The history of quantum mechanics is a fundamental part of the history of modern physics. The major chapters of this history begin with the emergence of quantum ideas to explain individual phenomena -- blackbody radiation, the photoelectric effect, solar emission spectra -- an era called the Old or Older quantum theories. The invention of wave mechanics by Schrodinger and expanded by many others triggers the "modern" era beginning around 1925 | https://huggingface.co/datasets/fmars/wiki_stem |
The relationship between mathematics and physics has been a subject of study of philosophers, mathematicians and physicists since Antiquity, and more recently also by historians and educators. Generally considered a relationship of great intimacy, mathematics has been described as "an essential tool for physics" and physics has been described as "a rich source of inspiration and insight in mathematics". In his work Physics, one of the topics treated by Aristotle is about how the study carried out by mathematicians differs from that carried out by physicists | https://huggingface.co/datasets/fmars/wiki_stem |
Criticism of the theory of relativity of Albert Einstein was mainly expressed in the early years after its publication in the early twentieth century, on scientific, pseudoscientific, philosophical, or ideological bases. Though some of these criticisms had the support of reputable scientists, Einstein's theory of relativity is now accepted by the scientific community. Reasons for criticism of the theory of relativity have included alternative theories, rejection of the abstract-mathematical method, and alleged errors of the theory | https://huggingface.co/datasets/fmars/wiki_stem |
In physics, the relativity of simultaneity is the concept that distant simultaneity – whether two spatially separated events occur at the same time – is not absolute, but depends on the observer's reference frame. This possibility was raised by mathematician Henri Poincaré in 1900, and thereafter became a central idea in the special theory of relativity.
Description
According to the special theory of relativity introduced by Albert Einstein, it is impossible to say in an absolute sense that two distinct events occur at the same time if those events are separated in space | https://huggingface.co/datasets/fmars/wiki_stem |
Ritz ballistic theory is a theory in physics, first published in 1908 by Swiss physicist Walther Ritz. In 1908, Ritz published Recherches critiques sur l'Électrodynamique générale, a lengthy criticism of Maxwell-Lorentz electromagnetic theory, in which he contended that the theory's connection with the luminiferous aether (see Lorentz ether theory) made it "essentially inappropriate to express the comprehensive laws for the propagation of electrodynamic actions. "
Ritz proposed a new equation, derived from the principles of the ballistic theory of electromagnetic waves, a theory competing with the special theory of relativity | https://huggingface.co/datasets/fmars/wiki_stem |
The Schiehallion experiment was an 18th-century experiment to determine the mean density of the Earth. Funded by a grant from the Royal Society, it was conducted in the summer of 1774 around the Scottish mountain of Schiehallion, Perthshire. The experiment involved measuring the tiny deflection of the vertical due to the gravitational attraction of a nearby mountain | https://huggingface.co/datasets/fmars/wiki_stem |
The history of scientific thought about the formation and evolution of the Solar System began with the Copernican Revolution. The first recorded use of the term "Solar System" dates from 1704. Since the seventeenth century, philosophers and scientists have been forming hypotheses concerning the origins of our Solar System and the Moon and attempting to predict how the Solar System would change in the future | https://huggingface.co/datasets/fmars/wiki_stem |
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