Submodule Test

Theorem
Let $\left({G, +, \circ}\right)_R$ be a unitary $R$-module.

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

Then $\left({H, +, \circ}\right)_R$ is a submodule of $G$ iff:


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

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

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