Module homomorphism

A map preserving addition and scalar multiplication between modules.

Module homomorphism

Let $M,N$ be left $R$ -modules . A module homomorphism is a function $f:M\to N$ such that for all $m,m'\in M$ and $r\in R$ ,

f(m+m')=f(m)+f(m') \quad\text{and}\quad f(rm)=r f(m).

Equivalently, $f$ is a group homomorphism on underlying additive groups that is $R$ -linear.

Module homomorphisms compose (see composition ), and their basic invariants are the kernel and image , which control exactness and quotients.

Examples:

For the $\mathbb Z$ -module $\mathbb Z$ , the map $f(n)=kn$ is a module homomorphism for any fixed $k\in\mathbb Z$ .
If $M=R^2$ and $N=R$ , the map $f(a,b)=a+rb$ (for fixed $r\in R$ ) is $R$ -linear.
(Edge case) A ring homomorphism need not be a module homomorphism unless one views the codomain as a module in the appropriate way.