Module Homomorphisms

Definition: $R$-Module Homomorphism

A map $\phi :M_1\to M_2$ of $R$-modules is an $R$-module homomorphism if it is a group homomorphism and it preserves the $R$-action. In other words,

1. $\phi (m_1+m_2)=\phi (m_1)+\phi (m_2)$
2. $\phi (r\cdot m)=r\cdot \phi (m)$

An $R$-module isomorphism is a bijective $R$-module homomorphism.

Note that not every group homomorphism is an $R$-module homomorphism!