A $A\_1$-$A\_2$- bimodule is on the left a unit M provided at the same time with a structure of module on a ring $A\_1$ and of a structure of module on the right on a ring $A\_2$ checking

$\backslash \; forall\; has\; \backslash \; in\; A\_1,\; \backslash \; quad\; \backslash \; forall\; X\; \backslash \; in\; M,\; \backslash \; quad\; \backslash \; forall\; B\; \backslash \; in\; A\_2,\; \backslash \; quad\; A.\; (x.b)\; =\; (a.x)\; .b$

Exemples

• All has - module on the right is also a $\backslash \; mathbb\; Z$- has - bimodule
• has is a has - has bimodule
• If has is commutative, all has - module can be seen as a has - has bimodule. More generally, for has unspecified, a has - module on the left can be seen like a $A-A^\; \{COp\}$ bimodule.

See too

• Multimodule