# Left Ideal is Left Module over Ring

## Theorem

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

Let $J \subseteq R$ be a left ideal of $R$.

Let $\circ : R \times J \to J$ be the restriction of $\times$ to $R \times J$.

Then $\struct {J, +, \circ}$ is a left module over $\struct {R, +, \times}$.

### Corollary

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

Then $\struct {R, +, \times}$ is a left module over $\struct {R, +, \times}$.

## Proof

By definition of a left ideal then $\circ$ is well-defined.

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

Follows directly from Ring Axiom $\text D$: Distributivity of Product over Addition

$\Box$

### Left Module Axiom $\text M 2$: (Right) Distributivity over Scalar Addition

Follows directly from Ring Axiom $\text D$: Distributivity of Product over Addition

$\Box$

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

Follows directly from Ring Axiom $\text M1$: Associativity of Product

$\blacksquare$