Hom module

A Hom module $\mathrm{Hom}_R(M,N)$ is the set of all module homomorphisms $f\colon M\to N$ between $R$-modules , equipped with pointwise addition $(f+g)(m)=f(m)+g(m)$. This makes $\mathrm{Hom}_R(M,N)$ an abelian group. If $R$ is a commutative ring , then $\mathrm{Hom}_R(M,N)$ is naturally an $R$-module via $(r\cdot f)(m)=r\,f(m)$.

In the noncommutative setting, additional module structures arise from bimodules: if $M$ is an $(R,S)$-bimodule and $N$ is a left $R$-module, then $\mathrm{Hom}_R(M,N)$ carries a natural left $S$-module structure by $(s\cdot f)(m)=f(ms)$.

Examples:

• For any left $R$-module $M$, evaluation at $1$ gives $\mathrm{Hom}_R(R,M)\cong M$.
• If $V,W$ are vector spaces over a field $k$, then $\mathrm{Hom}_k(V,W)$ is a vector space, naturally isomorphic to the space of $k$-linear maps $V\to W$.