Subring_Module_v2

Let $\struct {R, +, \times}$ be a ring. Let $\struct {S, +_S, \times_S}$ be a subring of $R$. Let $\struct {G, +_G, \circ}_R$ be an $R$-module. Let $\circ_S$ be the restriction of $\circ$ to $S \times G$. Then $\struct {G, +_G, \circ_S}_S$ is an $S$-module. The module $\struct {G, +_G, \circ_S}_S$ is called the '''$S$-module obtained from $\struct {G, +_G, \circ}_R$ by restricting scalar multiplication'''. If $\struct {G, +_G, \circ}_R$ is a unitary $R$-module and $1_R \in S$, then $\struct{G, +_G, \circ_S}_S$ is also unitary.