Module homomorphism and isomorphism, kernel, image and cokernel

Depends on

• Unital left and right modules over a ring; unqualified module means left module
• Quotient module $M/N$ with scalar multiplication on additive cosets
• Injection, surjection, bijection

Used by

• A ring with finitely many ideals of zero intersection whose quotients are Noetherian rings is Noetherian Corollary
• Localisation commutes with kernels images and cokernels Corollary
• An R-linear action of G on a left R-module, and a G-module over R Definition
• Exact sequences and short exact sequences of modules Definition
• Finitely presented modules and finitely presented algebras Definition
• Injective modules and the extension property Definition
• Invariant basis number and the rank of a free module Definition
• Projective modules and the lifting property Definition
• The abelian group Hom_R(M,N) and maps induced by pre- and postcomposition Definition
• Assuming the Axiom of Choice, the Chinese-remainder map Z/6Z -> Z/2Z direct-sum Z/3Z is an isomorphism by local tests Example
• Hom_ℤ(ℤ/m,ℤ/n)≅ℤ/gcd(m,n) for n≥1 Example
• When the group order is invertible the Reynolds operator retracts a ring onto its invariants Example
• A subring that admits a module retraction from a Noetherian ring is Noetherian Lemma
• Coextension of scalars Hom_R(S,M) carries its canonical left S-module structure Lemma
• Injective module maps remain injective after localisation Lemma
• Over a commutative ring the homomorphism group Hom_R(M,N) is an R-module Lemma
• Surjective module maps remain surjective after localisation Lemma
• Abelian groups and ℤ-modules have the same objects and morphisms Proposition
• Left modules over a fixed ring and module homomorphisms form the large locally small category R-Mod Proposition
• Module endomorphisms form a ring under pointwise addition and composition Proposition
• Module homomorphisms induce tensor-product homomorphisms functorially Proposition
• The canonical map M→ M/N is a surjective module homomorphism with kernel N; thus every submodule is a kernel Proposition
• A module homomorphism vanishing on N factors uniquely through M/N Theorem
• Abelian groups form an abelian category Theorem
• Additive functors from a ring to abelian groups are left modules Theorem
• Assuming the Axiom of Choice, a sequence of modules is exact exactly when all prime localisations are exact Theorem
• Assuming the Axiom of Choice, local criteria for zero modules and for injective, surjective, and bijective maps Theorem
• First isomorphism theorem for modules: M/ker f congimf Theorem
• For a commutative ring R, R-linear G-actions are exactly the compatible left R[G]-module structures Theorem
• For every ring R, the category R-Mod is complete and cocomplete Theorem
• Inverse limits preserve kernels Theorem
• Kernels and images of module homomorphisms are submodules, and injectivity is equivalent to trivial kernel Theorem
• Modules over a ring form an abelian category Theorem
• The Snake Lemma for modules Theorem
• Universal property of a direct sum of modules Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• McGerty, Algebra II: Rings and Modules, Section 3 (standard reference, not scraped)