Definition:Submodule

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 $$\left({S, +: \circ}\right)_R$$.