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 $\struct {R, +, \times}$.

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

$(M1)$ : Scalar Multiplication (Left) Distributes over Module Addition
Follows directly from ring axiom $(D)$: Product is Distributive over Addition

$(M2)$ : Scalar Multiplication (Right) Distributes over Scalar Addition
Follows directly from ring axiom $(D)$: Product is Distributive over Addition

$(M3)$ : Associativity of Scalar Multiplication
Follows directly from ring axiom $(M1)$: Associativity of Product

Also see

 * Leigh.Samphier/Sandbox/Right Ideal is Right Module over Ring