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 $$x \in H \Longrightarrow -x = \left({- 1_R}\right) \circ x \in H$$.

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