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

### Corollary

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

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

Then:

 $\ds \forall x, y \in R: \forall j \in J: \,$ $\ds \paren {x \circ_l j} \circ_r y$ $=$ $\ds \paren {x \times j} \times y$ Definition of $\circ_l$ and $\circ_r$ $\ds$ $=$ $\ds x \times \paren {j \times y}$ Ring Axiom $\text M1$: Associativity of Product $\ds$ $=$ $\ds x \circ_l \paren {j \circ_r y}$ Definition of $\circ_l$ and $\circ_r$

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

$\blacksquare$