Submodule Test

Theorem
Let $\struct {R, +_R, \times_R}$ be a ring with unity $1_R$.

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 $G$ :

Necessary Condition
Let $\struct {H, +, \circ}_R$ fulfil the conditions $\text {SM} 1$ and $\text {SM} 2$.

We have that $H \subseteq G$ such that $H \ne \O$.

We have from $\text {SM} 1$:
 * $\forall x, y \in H: x + y \in H$

We have that $\struct {R, +_R, \times_R}$ is a ring with unity whose unity is $1_R$.

Hence:

It follows from the Two-Step Subgroup Test that $H$ is a subgroup of $\struct {G, +}$.

It remains to be shown that $\struct {H, +, \circ}_R$ fulfils the module axioms as follows:



Hence $\struct {H, +, \circ}_R$ fulfils.



Hence $\struct {H, +, \circ}_R$ fulfils.



Hence $\struct {H, +, \circ}_R$ fulfils.

Sufficient Condition
Let $\struct {H, +, \circ}_R$ be a submodule of $G$.

As $H$ is an $R$-module, $\struct {H, +}$ is an abelian group.

As $\struct {H, +}$ is a group, it is closed under $+$ it follows that
 * $\forall x, y \in H: x + y \in H$

which is $\text {SM} 1$.

As $H$ is an $R$-module, it is closed for scalar product:
 * $\forall \lambda \in R, x \in H: \lambda \circ x \in H$

which is $\text {SM} 2$.