Definition:Submodule

Definition

Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {S, +, \circ}_R$ be an $R$-algebraic structure with one operation.

Let $T$ be a closed subset of $S$.

Let $\struct {T, +_T, \circ_T}_R$ be an $R$-module where:

$+_T$ is the restriction of $+$ to $T \times T$

$\circ_T$ is the restriction of $\circ$ to $R \times T$.

Then $\struct {T, +_T, \circ_T}_R$ is a submodule of $\struct {S, +, \circ}_R$.