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
• Exact finite tensor functors have projective right-module kernels Corollary
• Exact module tensor functors correspond to right-flat bimodules Corollary
• Finite one-sided exactness is equivalent to the existence of the corresponding adjoint Corollary
• Localisation commutes with kernels images and cokernels Corollary
• Upper semicontinuity of fibre cohomology dimensions 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
• The coinvariant functors Φₐ and the transition maps T Definition
• The Morita bicategory of rings and bimodules Definition
• A finite right exact functor needs no infinite-coproduct hypothesis Example
• A nonsplit simple has Hom-pairing diagonal two Example
• Assuming the Axiom of Choice, the Chinese-remainder map Z/6Z -> Z/2Z direct-sum Z/3Z is an isomorphism by local tests Example
• Fitting ideals of a diagonal two-by-two presentation Example
• Hom_ℤ(ℤ/m,ℤ/n)≅ℤ/gcd(m,n) for n≥1 Example
• The algebra of a two-object discrete groupoid is weak Hopf Example
• The dual-numbers tensor functor is right exact but not left exact Example
• The matrix-ring Morita pair with explicit tensor inverses 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
• Bilinear right exact functors are determined by their value on the regular modules Lemma
• Closed immersions are affine quotients and survive base change Lemma
• Coextension of scalars Hom_R(S,M) carries its canonical left S-module structure Lemma
• Coprime tangent cones force a power of the maximal ideal into the local ideal Lemma
• Degreewise direct sums and homogeneous free covers in graded modules Lemma
• Elementary ideals are independent of the presentation Lemma
• Finite-group invariants are exact when the group order is invertible Lemma
• Finite-support families of finite-dimensional vector spaces are locally finite but not finite Lemma
• Generation in bounded degree presents an FI-module as a quotient of free FI-modules Lemma
• Graded modules with degree-zero maps form an abelian category Lemma
• Graded tensor functors are k-linear, right exact, coproduct preserving and shift-coherent Lemma
• Homogeneous free presentations prove the graded comparison is an isomorphism Lemma
• Injective module maps remain injective after localisation Lemma
• Laurent-monomial decomposition of the projective Cech complex Lemma
• Over a commutative ring the homomorphism group Hom_R(M,N) is an R-module Lemma

…and 37 more results.

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)