Submodule Test

Theorem
Let $\struct {G, +, \circ}_R$ be a unitary $R$-module.

Let $H$ be a non-empty subset of $G$.

Then $\struct {H, +, \circ}_R$ is a submodule of $G$ :


 * $\forall x, y \in H: \forall \lambda \in R: x + y \in H, \lambda \circ x \in H$

Proof
If the conditions are fulfilled, then:
 * $x \in H \implies -x = \paren {-1_R} \circ x \in H$

Thus $H$ is a subgroup of $\struct {G, +}$ by the Two-Step Subgroup Test, and hence a submodule.