path stringlengths 11 71 | content stringlengths 75 124k |
|---|---|
RingTheory\FreeRing.lean | /-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin
-/
import Mathlib.GroupTheory.FreeAbelianGroup
/-!
# Free rings
The theory of the free ring over a type.
## Main definitions
* `FreeRing α` : the free (not com... |
RingTheory\Generators.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.Ideal.Cotangent
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.MvPolynomial.Tower
import Mathlib.RingTheory.TensorPro... |
RingTheory\Henselian.lean | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.Taylor
import Mathlib.RingTheory.LocalRing.ResidueField.Basic
import Mathlib.RingTheory.AdicCompletion.Basic
/-!
# Henselian rings
... |
RingTheory\HopfAlgebra.lean | /-
Copyright (c) 2024 Ali Ramsey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ali Ramsey
-/
import Mathlib.RingTheory.Bialgebra.Basic
/-!
# Hopf algebras
In this file we define `HopfAlgebra`, and provide instances for:
* Commutative semirings: `CommSemiring.toHop... |
RingTheory\Idempotents.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.Algebra.GeomSum
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Nilpotent.Defs
/-!... |
RingTheory\IntegralDomain.lean | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Chris Hughes
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.GroupTheory.SpecificGroups.Cyclic
/-!
# Integral domains
Assorte... |
RingTheory\IsAdjoinRoot.lean | /-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
/-!
# A predicate on adjoining root... |
RingTheory\IsTensorProduct.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.TensorProduct.Basic
import Mathlib.Algebra.Module.ULift
/-!
# The characteristic predicate of tensor product
## Main definitions
- `IsTensorPro... |
RingTheory\Jacobson.lean | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.JacobsonIdeal
/-!
# Jacobson Rings
The following conditions ... |
RingTheory\JacobsonIdeal.lean | /-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma
-/
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
/-!
# Jacobson radical
The Jacob... |
RingTheory\LaurentSeries.lean | /-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
import Mathlib.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.DedekindDomain.Bas... |
RingTheory\LittleWedderburn.lean | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Rodriguez
-/
import Mathlib.GroupTheory.ClassEquation
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
/-!
# We... |
RingTheory\LocalProperties.lean | /-
Copyright (c) 2021 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.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization... |
RingTheory\MatrixAlgebra.lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Eric Wieser
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.RingTheory.TensorProduct.Basic
/-!
We show `Matrix n n A ≃ₐ[R] (A ⊗[R] Matrix n n R)`.
-/
suppress_comp... |
RingTheory\MaximalSpectrum.lean | /-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.RingTheory.PrimeSpectrum
import Mathlib.RingTheory.Localization.AsSubring
/-!
# Maximal spectrum of a commutative ring
The maxima... |
RingTheory\Multiplicity.lean | /-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated.Basic
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.P... |
RingTheory\Nakayama.lean | /-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.RingTheory.JacobsonIdeal
/-!
# Nakayama's lemma
This file contains some alternative statements of Nakayama's Lemma as found in
[Stacks: Nakayama's Lemma]... |
RingTheory\Noetherian.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.Order.Filter.EventuallyConst
import Mathlib.Order.PartialSups
import Mathlib.Algebra.Module.Submodule.IterateMapComap
imp... |
RingTheory\NormTrace.lean | /-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Norm.Defs
import Mathlib.RingTheory.Trace.Defs
/-!
# Relation between Norms and Traces
-/
lemma Algebra.norm_one_add_smul {A B} [CommRing A] [C... |
RingTheory\Nullstellensatz.lean | /-
Copyright (c) 2021 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.RingTheory.Jacobson
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.MvPolynomial
import Mathlib.RingTheory.PrimeSpectrum
/-!
# Nullste... |
RingTheory\OrzechProperty.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.RingTheory.Finiteness
import Mathlib.Logic.Equiv.TransferInstance
/-!
# Orzech property of rings
In this file we define the following property of rings:
- `OrzechP... |
RingTheory\Perfection.lean | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Pi
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.CharP.Subring
import Mathlib.Algebra.Ring.Pi
import Mathlib.Analysis.SpecialFunctio... |
RingTheory\PiTensorProduct.lean | /-
Copyright (c) 2024 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.LinearAlgebra.PiTensorProduct
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Data.Finset.NoncommProd
/-!
# T... |
RingTheory\PolynomialAlgebra.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.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
/-!
# Algebr... |
RingTheory\PowerBasis.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.FieldTheory.Minpoly.Field
import Mathlib.LinearAlgebra.SModEq
/-!
# Power basis
This file defines a structure `PowerBasis R S`, giving a basis of the
`R`-a... |
RingTheory\Presentation.lean | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jung Tao Cheng, Christian Merten, Andrew Yang
-/
import Mathlib.LinearAlgebra.TensorProduct.RightExactness
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Gene... |
RingTheory\Prime.lean | /-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Associated.Basic
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Order.Group... |
RingTheory\PrimeSpectrum.lean | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Filippo A. E. Nuccio, Andrew Yang
-/
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib... |
RingTheory\PrincipalIdealDomain.lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.RingTheory.Ideal.Colon
import Mathlib.RingTheory.UniqueFactorizationDomain
/-!
# Principal i... |
RingTheory\QuotientNilpotent.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.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.Ideal.QuotientOperations
/-!
# Nilpotent elements in quotient rings
-/
theorem Ideal.isRadical_iff_qu... |
RingTheory\QuotientNoetherian.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.Noetherian
import Mathlib.RingTheory.Ideal.QuotientOperations
/-!
# Noetherian quotient rings and quotient modules
-/
instance Ideal.Quotient.is... |
RingTheory\QuotSMulTop.lean | /-
Copyright (c) 2024 Brendan Murphy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Brendan Murphy
-/
import Mathlib.LinearAlgebra.TensorProduct.RightExactness
/-!
# Reducing a module modulo an element of the ring
Given a commutative ring `R` and an element `r : R`,... |
RingTheory\ReesAlgebra.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.FiniteType
/-!
# Rees algebra
The Rees algebra of an ideal `I` is the subalgebra `R[It]` of `R[t]` defined as `R[It] = ⨁ₙ Iⁿ tⁿ`.
This is used ... |
RingTheory\RingHomProperties.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.Algebra.Category.Ring.Constructions
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.CategoryTheory.Iso
import Mathlib.RingTheory.Localization.Aw... |
RingTheory\RingInvo.lean | /-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kenny Lau
-/
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.Opposite
/-!
# Ring involutions
This file defines a ring involution as a structure exte... |
RingTheory\SimpleModule.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.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
... |
RingTheory\Support.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.PrimeSpectrum
import Mathlib.Algebra.Module.LocalizedModule
import Mathlib.RingTheory.Localization.AtPrime
/-!
# Support of a module
## Main re... |
RingTheory\SurjectiveOnStalks.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.Localization.AtPrime
import Mathlib.RingTheory.LocalRing.RingHom.Basic
import Mathlib.RingTheory.TensorProduct.Basic
/-!
# Ring Homomorphisms sur... |
RingTheory\UniqueFactorizationDomain.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.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... |
RingTheory\ZMod.lean | /-
Copyright (c) 2022 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best
-/
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
/-!
# Ri... |
RingTheory\AdicCompletion\Algebra.lean | /-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.Algebra.Module.Torsion
/-!
# Algebra instance on adic completion
In ... |
RingTheory\AdicCompletion\AsTensorProduct.lean | /-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
import Mathlib.Linea... |
RingTheory\AdicCompletion\Basic.lean | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Judith Ludwig, Christian Merten
-/
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.JacobsonIdeal
/-!
# Completion of a module with r... |
RingTheory\AdicCompletion\Exactness.lean | /-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.AdicCompletion.Functoriality
import Mathlib.RingTheory.Filtration
/-!
# Exac... |
RingTheory\AdicCompletion\Functoriality.lean | /-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.RingTheory.AdicCompletion.Algebra
import Mathlib.Algebra.DirectSum.Bas... |
RingTheory\Adjoin\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.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis
impo... |
RingTheory\Adjoin\FG.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.EuclideanDomain.Int
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
... |
RingTheory\Adjoin\Field.lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.AdjoinRoot
/-!
# Adjoining elements to a field
Some lemmas on ... |
RingTheory\Adjoin\PowerBasis.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.Adjoin.Basic
import Mathlib.RingTheory.PowerBasis
import Mathlib.LinearAlgebra.Matrix.Basis
/-!
# Power basis for `Algebra.adjoin R {x}`
This fi... |
RingTheory\Adjoin\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
-/
import Mathlib.RingTheory.Adjoin.FG
/-!
# Adjoining elements and being finitely generated in an algebra tower
## Main results
* `Algebra.fg_trans'`: if `S` is finitely gene... |
RingTheory\Bialgebra\Basic.lean | /-
Copyright (c) 2024 Ali Ramsey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ali Ramsey, Kevin Buzzard
-/
import Mathlib.RingTheory.Coalgebra.Basic
import Mathlib.RingTheory.TensorProduct.Basic
/-!
# Bialgebras
In this file we define `Bialgebra`s.
## Main defini... |
RingTheory\Bialgebra\Equiv.lean | /-
Copyright (c) 2024 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.RingTheory.Coalgebra.Equiv
import Mathlib.RingTheory.Bialgebra.Hom
/-!
# Isomorphisms of `R`-bialgebras
This file defines bundled isomorphisms ... |
RingTheory\Bialgebra\Hom.lean | /-
Copyright (c) 2024 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov, Amelia Livingston
-/
import Mathlib.RingTheory.Coalgebra.Hom
import Mathlib.RingTheory.Bialgebra.Basic
/-!
# Homomorphisms of `R`-bialgebras
This file ... |
RingTheory\Coalgebra\Basic.lean | /-
Copyright (c) 2023 Ali Ramsey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ali Ramsey, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.LinearAlgebra.TensorProduct.Finiten... |
RingTheory\Coalgebra\Equiv.lean | /-
Copyright (c) 2024 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.RingTheory.Coalgebra.Hom
/-!
# Isomorphisms of `R`-coalgebras
This file defines bundled isomorphisms of `R`-coalgebras. We simply mimic the ear... |
RingTheory\Coalgebra\Hom.lean | /-
Copyright (c) 2024 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov, Amelia Livingston
-/
import Mathlib.RingTheory.Coalgebra.Basic
/-!
# Homomorphisms of `R`-coalgebras
This file defines bundled homomorphisms of `R`-coa... |
RingTheory\Congruence\Basic.lean | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.GroupTheory.Congruence.Basic
/-!
# Congrue... |
RingTheory\Congruence\BigOperators.lean | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.GroupTheory.Congruence.BigOperators
import Mathlib.RingTheory.Congruence.Basic
/-!
# Interactions between `∑, ∏` and `RingCon`
-/
namespace RingCon
/-- C... |
RingTheory\Congruence\Opposite.lean | /-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.RingTheory.Congruence.Basic
import Mathlib.GroupTheory.Congruence.Opposite
/-!
# Congruences on the opposite ring
This file defines the order isomorphism ... |
RingTheory\Coprime\Basic.lean | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupWithZero.Action.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import... |
RingTheory\Coprime\Ideal.lean | /-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.RingTheory.Ideal.Operations
/-!
# An additional lemma about coprime ideals
This lemma generalis... |
RingTheory\Coprime\Lemmas.lean | /-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-!
# Add... |
RingTheory\DedekindDomain\AdicValuation.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: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valu... |
RingTheory\DedekindDomain\Basic.lean | /-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Polynomial.RationalRoot
/-!
# Dedekind rings and domains
This file... |
RingTheory\DedekindDomain\Different.lean | /-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Discriminant
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.NumberTheory.K... |
RingTheory\DedekindDomain\Dvr.lean | /-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Localization.LocalizationLocalization
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.Rin... |
RingTheory\DedekindDomain\Factorization.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: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
/-!
# Factorization of ideals and fractional ideals of Dedekind domains
Every nonzero ... |
RingTheory\DedekindDomain\FiniteAdeleRing.lean | /-
Copyright (c) 2023 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.AdicValuation
import Mathlib.RingTheory.DedekindDomain.Factorization
import Mathlib.Algebra.O... |
RingTheory\DedekindDomain\Ideal.lean | /-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.RingTheory.MaximalSpectrum
import Mathlib.RingTheory.ChainOfDivi... |
RingTheory\DedekindDomain\IntegralClosure.lean | /-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Bilin... |
RingTheory\DedekindDomain\PID.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.DedekindDomain.Dvr
import Mathlib.RingTheory.DedekindDomain.Ideal
/-!
# Criteria under which a Dedekind domain is a PID
This file contains some ... |
RingTheory\DedekindDomain\SelmerGroup.lean | /-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Data.ZMod.Quotient
import Mathlib.RingTheory.DedekindDomain.AdicValuation
/-!
# Selmer... |
RingTheory\DedekindDomain\SInteger.lean | /-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.RingTheory.DedekindDomain.AdicValuation
/-!
# `S`-integers and `S`-units of fraction fields of Dedekind domains
Let `K` be the fi... |
RingTheory\Derivation\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.Adjoin.Basic
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
/-!
# Derivations
... |
RingTheory\Derivation\Lie.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.Algebra.Lie.OfAssociative
import Mathlib.RingTheory.Derivation.Basic
/-!
# Results
- `Derivation.instLieAlgebra`: The `R`-derivation... |
RingTheory\Derivation\ToSquareZero.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.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
/-!
# Results
- `derivationToSquareZeroOfLift`: The `... |
RingTheory\DiscreteValuationRing\Basic.lean | /-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.RingT... |
RingTheory\DiscreteValuationRing\TFAE.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.Cotangent
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Valuation.ValuationRing
import Mathlib.RingTheory.Nakayam... |
RingTheory\Etale\Basic.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.QuotientNilpotent
import Mathlib.RingTheory.Smooth.Basic
import Mathlib.RingTheory.Unramified.Basic
/-!
# Etale morphisms
An `R`-algebra `A` is... |
RingTheory\Flat\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.RingTheory.Flat.Stability
import Mathlib.LinearAlgebra.TensorProduct.Tower
/-!
# Flat algebras and flat ring homomorphisms
We define flat algebra... |
RingTheory\Flat\Basic.lean | /-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Jujian Zhang
-/
import Mathlib.RingTheory.Noetherian
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.Algebra.Module.Proj... |
RingTheory\Flat\CategoryTheory.lean | /-
Copyright (c) 2024 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.RingTheory.Flat.Basic
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
/-!
# Tensoring with... |
RingTheory\Flat\EquationalCriterion.lean | /-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.RingTheory.Flat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Vanishing
import Mathlib.Algebra.Module.FinitePresentation
/-! # The equational criterion... |
RingTheory\Flat\Stability.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.RingTheory.Flat.Basic
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.Localiza... |
RingTheory\FractionalIdeal\Basic.lean | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
/-!
# Fractional ideals
This file defines fractional... |
RingTheory\FractionalIdeal\Norm.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.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
/-!
# Fractional ideal norms
This file defines the absolute ideal norm of a frac... |
RingTheory\FractionalIdeal\Operations.lean | /-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic
/-!
# More operations on fractional ideals... |
RingTheory\GradedAlgebra\Basic.lean | /-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Kevin Buzzard, Jujian Zhang, Fangming Li
-/
import Mathlib.Algebra.DirectSum.Algebra
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algebra.DirectSum.Internal
... |
RingTheory\GradedAlgebra\HomogeneousIdeal.lean | /-
Copyright (c) 2021 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic... |
RingTheory\GradedAlgebra\HomogeneousLocalization.lean | /-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.GradedAlgebra.Basic
/-!
# Homogeneous Localization
## Notation
- `ι` is a commutat... |
RingTheory\GradedAlgebra\Noetherian.lean | /-
Copyright (c) 2023 Fangming Li. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fangming Li
-/
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.GradedAlgebra.Basic
/-!
# The properties of a graded Noetherian ring.
This file proves that the 0-th grade ... |
RingTheory\GradedAlgebra\Radical.lean | /-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
/-!
This file contains a proof that the radical of any homogeneous ideal is a homogeneous ideal
#... |
RingTheory\HahnSeries\Addition.lean | /-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.RingTheory.HahnSeries.Basic
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Module.LinearMap.Defs
/-!
# Additive properties of Hahn series
... |
RingTheory\HahnSeries\Basic.lean | /-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Group.Support
import Mathlib.Order.WellFoundedSet
/-!
# Hahn Series
If `Γ` is ordered and `R` has zero, then `HahnSeries Γ R` consists of form... |
RingTheory\HahnSeries\Multiplication.lean | /-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Scott Carnahan
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Data.Finset.MulAntidiagonal
import Mathlib.Data.Finset.SMulAntidiagonal
import Mathlib.... |
RingTheory\HahnSeries\PowerSeries.lean | /-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.Data.Finsupp.PWO
/-!
# Comparison between Hahn series ... |
RingTheory\HahnSeries\Summable.lean | /-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.RingTheory.HahnSeries.Multiplication
/-!
# Summable families of Hahn Series
We introduce a notion of forma... |
RingTheory\HahnSeries\Valuation.lean | /-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.Valuation.Basic
/-!
# Valuations on Hahn Series rings
If `Γ` is a `LinearOrderedCancelA... |
RingTheory\Ideal\AssociatedPrime.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.LinearAlgebra.Span
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Noetherian
/-!
# ... |
RingTheory\Ideal\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.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
import... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.