Submodule Test

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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.

$\blacksquare$


Sources