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Module homomorphism

A map preserving addition and scalar multiplication between modules.

Let RR be a ring and let M,NM,N be left RR-. A module homomorphism is a f:M→Nf:M\to N such that for all m,m′∈Mm,m'\in M and r∈Rr\in R,

f(m+m′)=f(m)+f(m′)andf(rm)=rf(m).f(m+m')=f(m)+f(m') \quad\text{and}\quad f(rm)=r f(m).
Equivalent characterizations

Equivalently, ff is a homomorphism of the underlying additive groups and commutes with multiplication by every scalar in RR.

Examples
  • For the Z\mathbb Z-module Z\mathbb Z, the map f(n)=knf(n)=kn is a module homomorphism for every fixed k∈Zk\in\mathbb Z.
  • If RR is commutative, then f:R2→Rf:R^2\to R, f(a,b)=a+rbf(a,b)=a+rb, is RR-linear for each fixed r∈Rr\in R.