Right Ideal is Right Module over Ring

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

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

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

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

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

$(RM1)$ : Scalar Multiplication (Right) Distributes over Module Addition
Follows directly from ring axiom $(D)$: Product is Distributive over Addition

$(RM2)$ : Scalar Multiplication (Left) Distributes over Scalar Addition
Follows directly from ring axiom $(D)$: Product is Distributive over Addition

$(RM3)$ : Associativity of Scalar Multiplication
Follows directly from ring axiom $(M1)$: Associativity of Product

Also see

 * Left Ideal is Left Module over Ring