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 $\mathsf{Pr} \infty \mathsf{fWiki}$ in order to provide a less unwieldy term.

Also see

• Subring Module is Module

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.3$