Definition:Subring Module

Definition
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 $S \times G$.

The $S$-module $\struct {G, +_G, \circ_S}_S$ is called the subring module induced by $S$.

Also known as
This is seen to be referred to in the literature as the $S$-module obtained from $\struct {G, +_G, \circ}_R$ by restricting scalar multiplication.

The term subring module was coined by in order to provide a less unwieldy term.

Also see

 * Subring Module is Module