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.
$\blacksquare$
Also see
Sources
- 2003: P.M. Cohn: Basic Algebra: Groups, Rings and Fields ... (previous) ... (next): Chapter $4$: Rings and Modules: $\S 4.1$: The Definitions Recalled