Definition:Submodule

Definition
Let $\left({R, +, \circ}\right)$ be a ring.

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

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

Let $\left({T, +_T, \circ_T}\right)_R$ be an $R$-module where:
 * $+_T$ is the restriction of $+$ to $T \times T$ and
 * $\circ_T$ is the restriction of $\circ$ to $R \times T$.

Then $\left({T, +_T, \circ_T}\right)_R$ is a submodule of $\left({S, +, \circ}\right)_R$.