DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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The $R$ -module ${Hom}_{R} (M, N)$ over a commutative ring

Definition

Let $R$ be a commutative ring and let $M, N$ be $R$ -modules. The abelian group ${Hom}_{R} (M, N)$ of module homomorphisms (The abelian group ${Hom}_{R} (M, N)$ and maps induced by pre- and postcomposition) becomes an $R$ -module under pointwise scalar multiplication

$(r f) (m) : = r f (m) .$

The function $r f$ is $R$ -linear because, for $s \in R$ ,

$(r f) (s m) = r s f (m) = s r f (m) = s (r f) (m),$

where the middle equality uses commutativity (Commutative ring). The remaining module laws hold pointwise in $N$ . This module is the internal Hom module and is denoted ${Hom}_{R} (M, N)$ .

Depends on

Used by

Dependency tree · two levels

6 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

Stacks Project, Section 10.12: Tensor products (standard reference, not scraped)

The RR-module Hom⁡R(M,N)HomR​(M,N) over a commutative ring