path stringlengths 11 71 | content stringlengths 75 124k |
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RingTheory\Ideal\Basis.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.LinearAlgebra.Basis
import Mathlib.RingTheory.Ideal.Basic
/-!
# The basis of ideals
Some results involving `Ideal` an... |
RingTheory\Ideal\Colon.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Operations
/-!
# The colon ideal
This file defines `Submodule.colon N P` as the ideal of all elements `r... |
RingTheory\Ideal\Cotangent.lean | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.Dimension.FreeAndStron... |
RingTheory\Ideal\IdempotentFG.lean | /-
Copyright (c) 2018 Mario Carneiro, Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Buzzard
-/
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Finiteness
import Mathlib.Order.Basic
/-!
## Lemmas on idempotent fin... |
RingTheory\Ideal\IsPrimary.lean | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
/-!
# Primary ideals
A proper ideal `I` is primary iff `xy ∈ I` implies `x ∈ I` or `y ∈ radical I`.
## Main definitions
- `Ideal.... |
RingTheory\Ideal\IsPrincipal.lean | /-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.RingTheory.PrincipalIdealDomain
/-!
# Principal Ideals
This file deals with the set of principal ideals of a `CommRing R`.
## Main definitions and res... |
RingTheory\Ideal\Maps.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
-/
assert_not_exists Basis -- See `RingTheory.Ideal.Basis`
assert_not_exists Submodule.hasQuotient... |
RingTheory\Ideal\MinimalPrime.lean | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Order.Minimal
/-!
# Minimal primes
We provide various results con... |
RingTheory\Ideal\Norm.lean | /-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Data.Finsupp.Fintype
import Mathlib.Data.Int.Absolut... |
RingTheory\Ideal\Operations.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
/-!
# More operations on modules and ideals
-/
assert_n... |
RingTheory\Ideal\Over.lean | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
/-!
# Ideals over/under ideals
This fil... |
RingTheory\Ideal\Pointwise.lean | /-
Copyright (c) 2024 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.RingTheory.Ideal.Maps
/-! # Pointwise instances on `Ideal`s
This file provides the action `Ideal.pointwiseMulAction` which morally matches the action of
`m... |
RingTheory\Ideal\Prod.lean | /-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.RingTheory.Ideal.Maps
/-!
# Ideals in product rings
For commutative rings `R` and `S` and ideals `I ≤ R`, `J ≤ S`, we define `Ideal.prod I J` as the
pr... |
RingTheory\Ideal\Quotient.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib... |
RingTheory\Ideal\QuotientOperations.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Patrick Massot
-/
import Mathlib.Algebra.Algebra.Subalgebra.Operations
import Mathlib.Algebra.Ring.Fin
import Mathlib.RingTheory.Ideal.Quotient
/-!
# More operations on modules ... |
RingTheory\Int\Basic.lean | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.RingTheory.PrincipalIdealDomain... |
RingTheory\IntegralClosure\IntegrallyClosed.lean | /-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.IntegralClosure.Algebra.Basic
import Mathlib.RingTheory.Localization.Integral
/-!
# Integrally closed rings
An integrally closed ring `R` contai... |
RingTheory\IntegralClosure\IntegralRestrict.lean | /-
Copyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
import Mathlib.RingTheory.LocalProperties
import Mathlib.RingTheory.Localization.NormTrace
import M... |
RingTheory\IntegralClosure\Algebra\Basic.lean | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.IntegralClosure.Algebra.Defs
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap
/-!
# ... |
RingTheory\IntegralClosure\Algebra\Defs.lean | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs
/-!
# Integral algebras
## Main definitions
Let `R` be a `CommRing` and let `A` be an R-algebra.
* `Algebra.IsInte... |
RingTheory\IntegralClosure\IsIntegral\Basic.lean | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.RingTheory.Polynomial.Tower
/-!
# Properties of integral elem... |
RingTheory\IntegralClosure\IsIntegral\Defs.lean | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Eval
/-!
# Integral closure of a subring.
If A is an R-algebra then `a : A` is integral over R if it is a root of a monic polynomial
with co... |
RingTheory\IntegralClosure\IsIntegralClosure\Basic.lean | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs
import Mathlib.RingTheory.IntegralClosure.Algebra.Basic
import Mathlib.RingTheory.FiniteType
import Mathlib.Rin... |
RingTheory\IntegralClosure\IsIntegralClosure\Defs.lean | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs
/-!
# Integral closure as a characteristic predicate
## Main definitions
Let `R` be a `CommRing` and let `A` be an ... |
RingTheory\Kaehler\Basic.lean | /-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... |
RingTheory\Kaehler\CotangentComplex.lean | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Kaehler.Polynomial
import Mathlib.RingTheory.Generators
/-!
# Naive cotangent complex associated to a presentation.
Given a presentation `0 → I... |
RingTheory\Kaehler\Polynomial.lean | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Kaehler.Basic
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.Derivation
/-!
# The Kaehler differential module of po... |
RingTheory\Localization\Algebra.lean | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.Algebra.Module.LocalizedModule
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingTheory.Localiza... |
RingTheory\Localization\AsSubring.lean | /-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Junyan Xu
-/
import Mathlib.RingTheory.Localization.LocalizationLocalization
/-!
# Localizations of domains as subalgebras of the fraction field.
Given a domain `A` with fra... |
RingTheory\Localization\AtPrime.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
import Mathl... |
RingTheory\Localization\BaseChange.lean | /-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Jujian Zhang
-/
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.RingTheory.Localization.Module
/-!
# Localized Module
Given a commutative semiring `R`, a multipli... |
RingTheory\Localization\Basic.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTh... |
RingTheory\Localization\Cardinality.lean | /-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.RingTheory.Artinian
/-!
# Cardinality of localizations
In this file, we establish the cardinality of locali... |
RingTheory\Localization\Finiteness.lean | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.Algebra.Module.LocalizedModuleIntegers
import Mathlib.RingTheory.Localization.Algebra
import Mathlib.RingTheory.LocalProperties
/-!
# Finiteness ... |
RingTheory\Localization\FractionRing.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.RingTheory.Localization.Basic
import Mathlib.Algebra.Field.E... |
RingTheory\Localization\Ideal.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Localization.Basic
/-!
# Ideals in... |
RingTheory\Localization\Integer.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.Basic
/-!
# Integer elements of a localization
## Main definitions
* ... |
RingTheory\Localization\Integral.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Grou... |
RingTheory\Localization\InvSubmonoid.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.GroupTheory.Submonoid.Inverses
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Loc... |
RingTheory\Localization\LocalizationLocalization.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingT... |
RingTheory\Localization\Module.lean | /-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu, Anne Baanen
-/
import Mathlib.LinearAlgebra.Basis
import Mathlib.Algebra.Module.LocalizedModule
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Lo... |
RingTheory\Localization\NormTrace.lean | /-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Norm.Basic
import Mathlib.RingTheory.Discriminant
/-!
# Field/algebra norm / trace and localizatio... |
RingTheory\Localization\NumDen.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
import Mathli... |
RingTheory\Localization\Submodule.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.... |
RingTheory\Localization\Away\AdjoinRoot.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.RingTheory.Localization.Away.Basic
/-!
The `R`-`AlgEquiv` b... |
RingTheory\Localization\Away\Basic.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.UniqueFactorizationDomain
import Mathlib.RingTheory.Localization.Basic
/-!
# Localiza... |
RingTheory\LocalRing\Basic.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.RingTheory.LocalRing.Defs
import Mathlib.RingTheory.Ideal.Basic
/-!
# Local rings
We prove basic properties of local rings.
-/
... |
RingTheory\LocalRing\Defs.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Ring.Defs
/-!
# Local rings
Define local rings as commutative rings having a unique m... |
RingTheory\LocalRing\Module.lean | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Nakayama
import Mathlib.Algebra.Module.FinitePresentation
import Mathlib.LinearAlgebra.FiniteDimensional
impo... |
RingTheory\LocalRing\MaximalIdeal\Basic.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Defs
import Mathlib.RingTheory.JacobsonIdeal
/-!
# Maximal ideal of local rings
We prove basic... |
RingTheory\LocalRing\MaximalIdeal\Defs.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.RingTheory.LocalRing.Basic
/-!
# Maximal ideal of local rings
We define the maximal ideal of a local ring as the ideal of all no... |
RingTheory\LocalRing\ResidueField\Basic.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.RingTheory.LocalRing.ResidueField.Defs
import Mathlib.RingTheory.LocalRing.RingHom.Basic
import Mathlib.RingTheory.LocalRing.Maxima... |
RingTheory\LocalRing\ResidueField\Defs.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
import Mathlib.RingTheory.Ideal.Quotient
/-!
# Residue Field of local rings
## Main defi... |
RingTheory\LocalRing\RingHom\Basic.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.RingTheory.LocalRing.RingHom.Defs
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Defs
import Mathlib.RingTheory.Ideal.Maps
import... |
RingTheory\LocalRing\RingHom\Defs.lean | /-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Ring.Hom.Defs
/-!
# Local rings homomorphisms
We Define local ring homomorhpisms.
##... |
RingTheory\MvPolynomial\Basic.lean | /-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.LinearAlgebra.FinsuppVectorS... |
RingTheory\MvPolynomial\Homogeneous.lean | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolyn... |
RingTheory\MvPolynomial\Ideal.lean | /-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Ideal
import Mathlib.Algebra.MvPolynomial.Division
/-!
# Lemmas about ideals of `MvPolynomial`
Notably this contains results about mo... |
RingTheory\MvPolynomial\Localization.lean | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.R... |
RingTheory\MvPolynomial\NewtonIdentities.lean | /-
Copyright (c) 2023 Michael Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Lee
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Data.Finset.Card
impo... |
RingTheory\MvPolynomial\Symmetric.lean | /-
Copyright (c) 2020 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang, Johan Commelin
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Combinatorics.Enumerative.Partition
/-!
# S... |
RingTheory\MvPolynomial\Tower.lean | /-
Copyright (c) 2022 Yuyang Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuyang Zhao
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.MvPolynomial.Basic
/-!
# Algebra towers for multivariate polynomial
This file proves some basic results about ... |
RingTheory\MvPolynomial\WeightedHomogeneous.lean | /-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPol... |
RingTheory\MvPowerSeries\Basic.lean | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Algebra.Order.Antidiag.Finsupp
import Mathlib.Data.Finsupp.Antidiagonal
import Mathlib.Data.Finsu... |
RingTheory\MvPowerSeries\Inverse.lean | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Group.Units
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors
import Mathlib.Rin... |
RingTheory\MvPowerSeries\NoZeroDivisors.lean | /-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.Data.Finsupp.WellFounded
/-! # ZeroDivisors in a MvPowerSeries ring
- `mem_nonZeroDiviso... |
RingTheory\MvPowerSeries\Trunc.lean | /-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.Data.Finsupp.Interval
/-!
# Formal (multivariate) power series - Truncation
`MvPowerSeries... |
RingTheory\Nilpotent\Basic.lean | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Associated.Basic
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Module.Defs
import Mathli... |
RingTheory\Nilpotent\Defs.lean | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Nat.Lattice
/-!
# Definition... |
RingTheory\Nilpotent\Lemmas.lean | /-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
/-!
# Nilpote... |
RingTheory\NonUnitalSubring\Basic.lean | /-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.GroupTheory.Subsemigroup.Center
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
/-!
# `NonUnitalSubring... |
RingTheory\NonUnitalSubsemiring\Basic.lean | /-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Subsemigroup.Membership
import Mathlib.Algebra.GroupWithZero.Center
import Mathlib.Algebr... |
RingTheory\Norm\Basic.lean | /-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.Norm.Defs
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Matrix.ToL... |
RingTheory\Norm\Defs.lean | /-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.LinearAlgebra.Determinant
/-!
# Norm for (finite) ring extensions
Suppose we have an `R`-algebra `S` with a finite basis. For each `s : S`,
the determinant... |
RingTheory\OreLocalization\Basic.lean | /-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.Algebra.Group.Submonoid.DistribMulAction
import Mathlib.RingTheory.OreLocalization.OreSet
/-!
# Localization over left... |
RingTheory\OreLocalization\OreSet.lean | /-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge
-/
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Algebra.Ring.Regular
/-!
# (Left) Ore sets
... |
RingTheory\OreLocalization\Ring.lean | /-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Field.Defs
import Math... |
RingTheory\Polynomial\Basic.lean | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTheory.Polynomial.Content
/-!
# ... |
RingTheory\Polynomial\Bernstein.lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Su... |
RingTheory\Polynomial\Chebyshev.lean | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
/-!
# Chebyshev polynomials
... |
RingTheory\Polynomial\Content.lean | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... |
RingTheory\Polynomial\Dickson.lean | /-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomia... |
RingTheory\Polynomial\GaussLemma.lean | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.Localization.Integral
import Mathlib.RingTheory.I... |
RingTheory\Polynomial\IntegralNormalization.lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polyno... |
RingTheory\Polynomial\IrreducibleRing.lean | /-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Nilpotent
/-!
# Polynomials over an irreducible ring
This file contains results about the polyn... |
RingTheory\Polynomial\Nilpotent.lean | /-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Uthaiwat, Oliver Nash
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.Algebra.Polynomial.Identities
import Mathlib.RingTheory.I... |
RingTheory\Polynomial\Opposites.lean | /-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
/-! # Interactions between `R[X]` and `Rᵐᵒᵖ[X]`
This file contains the basic API for "pushing through" the isom... |
RingTheory\Polynomial\Pochhammer.lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.... |
RingTheory\Polynomial\Quotient.lean | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, David Kurniadi Angdinata, Devon Tuma, Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Quotient... |
RingTheory\Polynomial\RationalRoot.lean | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
import Mathlib.RingTheory.Localization.NumDen
import Mathlib.RingTheory.Polynomial.ScaleRoots
/-!
# Rational roo... |
RingTheory\Polynomial\ScaleRoots.lean | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... |
RingTheory\Polynomial\Selmer.lean | /-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Analysis.Complex.Polynomial.UnitTrinomial
import Mathlib.RingTheory.Polynomial.GaussLemma
import Mathlib.Tactic.LinearCombination
/-!
# Irreducibili... |
RingTheory\Polynomial\SeparableDegree.lean | /-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.FieldTheory.Separable
/-!
# Separable degree
This file contains basics ab... |
RingTheory\Polynomial\Tower.lean | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yuyang Zhao
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Polynomial.AlgebraMap
/-!
# Algebra towers for polynomial
This file proves some basic results about t... |
RingTheory\Polynomial\Vieta.lean | /-
Copyright (c) 2020 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.MvPolynomial.Symmetric
/-!
# Vieta's Formula
The main result is `Multiset.prod_X_add_C_eq_sum_esymm... |
RingTheory\Polynomial\Wronskian.lean | /-
Copyright (c) 2024 Jineon Back and Seewoo Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jineon Baek, Seewoo Lee
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.LinearAlgebra.SesquilinearForm
/-!
# Wr... |
RingTheory\Polynomial\Cyclotomic\Basic.lean | /-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import... |
RingTheory\Polynomial\Cyclotomic\Eval.lean | /-
Copyright (c) 2021 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.Pad... |
RingTheory\Polynomial\Cyclotomic\Expand.lean | /-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Data.ZMod.Algebra
/-!
# Cyclotomic polynomials and `expand`.
We gather results relating cyclo... |
RingTheory\Polynomial\Cyclotomic\Roots.lean | /-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
import Mathlib.RingTheory.RootsOfUnity.Minpoly
/-!
# Roots of cyclotomic polynomials.
We gather results abou... |
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