DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13

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The abelian group ${Hom}_{R} (M, N)$ and maps induced by pre- and postcomposition

Definition

For left $R$ -modules $M, N$ , the set ${Hom}_{R} (M, N)$ of module homomorphisms is an abelian group under pointwise addition, with zero the zero homomorphism and inverse $(- f) (m) = - f (m)$ (Module homomorphism and isomorphism, kernel, image and cokernel, Group and abelian group).

For a homomorphism $u : M \to N$ and any module $X$ , postcomposition and precomposition give homomorphisms $u_{*} : {Hom}_{R} (X, M) \to {Hom}_{R} (X, N), f \mapsto u \circ f,$ $u^{*} : {Hom}_{R} (N, X) \to {Hom}_{R} (M, X), g \mapsto g \circ u .$ Composition is associative, so $(v \circ u)_{*} = v_{*} \circ u_{*}$ and $(v \circ u)^{*} = u^{*} \circ v^{*}$ ; identity maps induce identity maps.

Depends on

Used by

Hom_ℤ(-,ℤ) need not preserve surjections on the right Counterexample
Coinduction sends injective abelian groups to injective modules Lemma
Covariant and contravariant Hom are left exact Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 17 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
The Stacks Project, Algebra (standard reference, not scraped)
P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)

The abelian group Hom⁡R(M,N)HomR​(M,N) and maps induced by pre- and postcomposition