An $R$-linear action of $G$ on a left $R$-module, and a $G$-module over $R$

Let $R$ be a commutative ring (Commutative ring), let $G$ be a group (Group and abelian group), and let $M$ be a left $R$-module (Unital left and right modules over a ring; unqualified module means left module).

A left $G$-action on $M$ by $R$-linear maps is a left group action $G\times M\to M$, $(g,m)\mapsto g\cdot m$, in the sense of Left group actions, transitive actions, and faithful actions, such that for every $g\in G$ the map $m⟼g\cdot m$ is an $R$-module homomorphism $M\to M$ (Module homomorphism and isomorphism, kernel, image and cokernel).

Equivalently, the action satisfies $g\cdot (m+m^{\mathrm{\prime }})=g\cdot m+g\cdot m^{\mathrm{\prime }},g\cdot (rm)=r(g\cdot m)$ for all $g\in G$, $m,m^{\mathrm{\prime }}\in M$, and $r\in R$.

An $R$-linear $G$-module, or simply a $G$-module over $R$, is a left $R$-module together with such an action.

• Because $g^{-1}$ acts as an inverse in the group-action sense, each map $m\mapsto g\cdot m$ is automatically an $R$-module automorphism.

• When $R$ is a field, this is the module-language form of a representation.