Ideal is Bimodule over Ring

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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:

\(\, \displaystyle \forall x, y \in R \land \forall j \in J: \, \) \(\displaystyle \paren {x \circ_l j} \circ_r y\) \(=\) \(\displaystyle \paren {x \times j} \times y\) Definition of $\circ_l$ and $\circ_r$
\(\displaystyle \) \(=\) \(\displaystyle x \times \paren c{j \times y}\) Ring axiom $(M \, 2)$ : Associativity of Product
\(\displaystyle \) \(=\) \(\displaystyle x \circ_l \paren {j \circ_r y}\) Definition of $\circ_l$ and $\circ_r$


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

$\blacksquare$


Also see


Sources