Ideal is Bimodule over Ring

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

Let $J \subseteq R$ be an ideal of $R$.

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

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

Then $\struct {J, +, \circ_l, \circ_r}$ is a bimodule over $\struct {R, +, \times}$.

Proof
By definition of an ideal, $J$ is both a left ideal and a right ideal.

From Left Ideal is Left Module over Ring then $\struct{J, +, \circ_l}$ is a left module.

From Right Ideal is Right Module over Ring then $\struct{J, +, \circ_r}$ is a right module.

By ring axiom $(M \, 1)$ : Associativity of Product then:

Hence $\struct {J, +, \circ_l, \circ_r}$ is a bimodule over $\struct {R, +, \times}$ By definition.

Also see

 * Ring is Bimodule over Ring