Module_of_All_Mappings_is_Module_v2

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G, \circ}_R$ be an $R$-module. Let $S$ be a set. Let $\struct {G^S, +_G', \circ}_R$ be the module of all mappings from $S$ to $G$. Then $\struct {G^S, +_G', \circ}_R$ is an $R$-module.