# Subring Module is Module

## Theorem

Let $\struct {R, +, \times}$ be a ring.

Let $\struct {S, +_S, \times_S}$ be a subring of $R$.

Let $\struct {G, +_G, \circ}_R$ be an $R$-module.

Let $\circ_S$ be the restriction of $\circ$ to $S \times G$.

Let $\struct {G, +_G, \circ_S}_S$ be subring module induced by $S$.

Then $\struct {G, +_G, \circ_S}_S$ is an $S$-module.

### Unitary Subring

Let $\struct {R, +, \times}$ be a ring with unity.

Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module.

Let $1_R \in S$.

Then $\struct{G, +_G, \circ_S}_S$ is also unitary.

### Special Case

Let $S$ be a subring of the ring $\struct {R, +, \circ}$.

Let $\circ_S$ be the restriction of $\circ$ to $S \times R$.

Then $\struct {R, +, \circ_S}_S$ is an $S$-module.

## Proof

We have that:

$\forall a, b \in S: a +_S b = a + b$
$\forall a, b \in S: a \times_S b = a \times b$
$\forall a \in S: \forall x \in G = a \circ_S x = a \circ x$

as $+_S$, $\times_S$ and $\circ_S$ are restrictions.

Let us verify the module axioms.

### Module Axiom $\text M 1$: Distributivity over Module Addition

We need to show that:

$\forall a \in S: \forall x, y \in G: a \circ_S \paren {x +_G y} = a \circ_S x +_G a \circ_S y$

We have:

 $\ds a \circ_S \paren {x +_G y}$ $=$ $\ds a \circ \paren {x +_G y}$ $\ds$ $=$ $\ds a \circ x +_G a \circ y$ Module Axiom $\text M 1$: Distributivity over Module Addition on $\struct {G, +_G, \circ}_R$ $\ds$ $=$ $\ds a \circ_S x +_G a \circ_S y$

$\Box$

### Module Axiom $\text M 2$: Distributivity over Scalar Addition

We need to show that:

$\forall a, b \in S: \forall x \in G: \paren {a +_S b} \circ_S x = a \circ_S x +_G b \circ_S y$

We have:

 $\ds \paren {a +_S b} \circ_S x$ $=$ $\ds \paren {a + b} \circ x$ $\ds$ $=$ $\ds a \circ x + b \circ x$ Module Axiom $\text M 2$: Distributivity over Scalar Addition on $\struct {G, +_G, \circ}_R$ $\ds$ $=$ $\ds a \circ_S x +_G b \circ_S x$

$\Box$

### Module Axiom $\text M 3$: Associativity

We need to show that:

$\forall a, b \in S: \forall x \in G: \paren {a \times_S b} \circ_S x = a \circ_S \paren {b \circ_S x}$

We have:

 $\ds \paren {a \times_S b} \circ_S x$ $=$ $\ds \paren {a \times b} \circ x$ $\ds$ $=$ $\ds a \circ \paren {b \circ x}$ Module Axiom $\text M 3$: Associativity on $\struct {G, +_G, \circ}_R$ $\ds$ $=$ $\ds a \circ_S \paren {b \circ_S x}$

$\Box$

Thus $\struct {G, +_G, \circ_S}_S$ is an $S$-module.

$\blacksquare$