Definition:Bimodule
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Definition
Let $\struct {R, +_R, \times_R}$ and $\struct {S, +_S, \times_S}$ be rings.
Let $\struct {G, +_G}$ be an abelian group.
Let $\circ_R : R \times G \to G$ and $\circ_S : G \times S \to G$ be binary operations such that:
- $(1): \quad \struct {G, +_G, \circ_R}$ is a left module
- $(2): \quad \struct {G, +_G, \circ_S}$ is a right module
- $(3): \quad \forall \lambda \in R: \forall \mu \in S: \forall x \in G: \paren {\lambda \circ_R x} \circ_S \mu = \lambda \circ_R \paren {x \circ_S \mu}$
Then $\struct {G, +_G, \circ_R, \circ_S}$ is a bimodule over $\tuple {R, S}$.
If $\struct {S, +_S, \times_S} = \struct {R, +_R, \times_R}$ then a bimodule over $\tuple {R, R}$ is simply called a bimodule over $R$
Also known as
A bimodule over $\tuple {R, S}$ can also be referred to as an $\tuple {R, S}$-bimodule.
Sources
- 2003: P.M. Cohn: Basic Algebra: Groups, Rings and Fields ... (previous) ... (next): Chapter $4$: Rings and Modules: $\S 4.1$: The Definitions Recalled