# 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 $G$ if and only if:

 $\text {(SM1)}: \quad$ $\ds \forall x, y \in H: \,$ $\ds x + y$ $\in$ $\ds H$ $\text {(SM2)}: \quad$ $\ds \forall x \in H: \forall \lambda \in R: \,$ $\ds \lambda \circ x$ $\in$ $\ds H$

## Proof

### Necessary Condition

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

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

Hence:

 $\ds x$ $\in$ $\ds H$ $\ds \leadsto \ \$ $\ds \paren {-1_R} \circ x$ $\in$ $\ds H$ from $\text {SM} 2$ $\ds \leadsto \ \$ $\ds -x$ $\in$ $\ds H$ Definition of Unity of Ring

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:

 $(\text M 1)$ $:$ Scalar Multiplication (Left) Distributes over Module Addition $\ds \forall \lambda \in R: \forall x, y \in H:$ $\ds \lambda \circ \paren {x + y}$ $\ds =$ $\ds \paren {\lambda \circ x} + \paren {\lambda \circ y}$ $(\text M 2)$ $:$ Scalar Multiplication (Right) Distributes over Scalar Addition $\ds \forall \lambda, \mu \in R: \forall x \in H:$ $\ds \paren {\lambda +_R \mu} \circ x$ $\ds =$ $\ds \paren {\lambda \circ x} + \paren {\mu \circ x}$ $(\text M 3)$ $:$ Associativity of Scalar Multiplication $\ds \forall \lambda, \mu \in R: \forall x \in H:$ $\ds \paren {\lambda \times_R \mu} \circ x$ $\ds =$ $\ds \lambda \circ \paren {\mu \circ x}$

Module Axiom $\text M 1$
 $\ds \forall x, y \in H: \,$ $\ds x + y$ $\in$ $\ds H$ from $\text {SM} 1$ $\ds \lambda \circ x$ $\in$ $\ds H$ from $\text {SM} 2$ $\ds \lambda \circ y$ $\in$ $\ds H$ from $\text {SM} 2$ $\ds \leadsto \ \$ $\ds \paren {\lambda \circ x} + \paren {\lambda \circ y}$ $\in$ $\ds H$ from $\text {SM} 1$ $\ds \leadsto \ \$ $\ds \paren {\lambda \circ x} + \paren {\lambda \circ y}$ $\in$ $\ds G$ as $H \subseteq G$ $\ds \leadsto \ \$ $\ds \paren {\lambda \circ x} + \paren {\lambda \circ y}$ $=$ $\ds \lambda \circ \paren {x + y}$ Module Axiom $\text M 1$: Distributivity over Module Addition as applied to $\struct {G, +, \circ}_R$

Hence $\struct {H, +, \circ}_R$ fulfils Module Axiom $\text M 1$: Distributivity over Module Addition.

Module Axiom $\text M 2$
 $\ds \forall x \in H: \forall \lambda \in R: \,$ $\ds \lambda \circ x$ $\in$ $\ds H$ from $\text {SM} 2$ $\ds \leadsto \ \$ $\ds \forall \mu \in R: \,$ $\ds \paren {\lambda +_R \mu} \circ x$ $\in$ $\ds H$ from $\text {SM} 2$: as $\lambda +_R \mu \in R$ $\ds \leadsto \ \$ $\ds \paren {\lambda +_R \mu} \circ x$ $\in$ $\ds G$ as $H \subseteq G$ $\ds \leadsto \ \$ $\ds \paren {\lambda +_R \mu} \circ x$ $=$ $\ds \paren {\lambda \circ x} + \paren {\mu \circ x}$ Module Axiom $\text M 2$: Distributivity over Scalar Addition as applied to $\struct {G, +, \circ}_R$

Hence $\struct {H, +, \circ}_R$ fulfils Module Axiom $\text M 2$: Distributivity over Scalar Addition.

Module Axiom $\text M 3$
Associativity
 $\ds \forall x \in H: \forall \lambda \in R: \,$ $\ds \lambda \circ x$ $\in$ $\ds H$ from $\text {SM} 2$ $\ds \leadsto \ \$ $\ds \forall \mu \in R: \,$ $\ds \paren {\lambda \times_R \mu} \circ x$ $\in$ $\ds H$ from $\text {SM} 2$: as $\lambda \times_R \mu \in R$ $\ds \leadsto \ \$ $\ds \paren {\lambda \times_R \mu} \circ x$ $\in$ $\ds G$ as $H \subseteq G$ $\ds \leadsto \ \$ $\ds \paren {\lambda \times_R \mu} \circ x$ $=$ $\ds \lambda \circ \paren {\mu \circ x}$ Module Axiom $\text M 3$: Associativity as applied to $\struct {G, +, \circ}_R$

Hence $\struct {H, +, \circ}_R$ fulfils Module Axiom $\text M 3$: Associativity.

### 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 $\text {SM} 2$.

$\blacksquare$