Bimodule

2020 Mathematics Subject Classification: Primary: 16D20 [MSN][ZBL]

double module

An Abelian group $B$ that is a left module over a ring $R$ and a right module over a ring $S$, and is such that $(rb)s = r(bs)$ for all $r\in R$, $b \in B$, $s \in S$. One writes ${}_R B_S$, or that $B$ is an $(R,S)$-bimodule. The bimodule $B$ may be regarded as a left $R \otimes S^{\mathrm{op}}$-module, where $S^{\mathrm{op}}$ is the opposite ring (dually isomorphic, anti-isomorphic) to $S$, while $\otimes$ denotes the tensor product over the ring of integers, and $(r\otimes s)b = rbs$. For every left $R$-module $M$ one has the situation ${}_R M_E$, where $E$ is the ring of endomorphisms of $M$. Any ring $R$ can be given the natural structure of an $(R,R)$-bimodule.

Comments

A bimodule morphism is a mapping from a bimodule ${}_R B_S$ into a bimodule ${}_R C_S$ that is left $R$-linear and right $S$-linear. The category of $(R,S)$-bimodules with bimodule morphisms is a Grothendieck category.

The centre of an $(R,R)$-bimodule (also called an $R$-bimodule) $B$ is defined to be the set $$Z_R(B) = \{x \in B : rx = xr \ \text{for all}\ r \in R\}\ .$$ Clearly $Z_R(B)$ is a two-sided $Z_R(R)$--module. In particular, when $R$ is commutative, the distinction between left and right modules disappears and any $R$-module may be regarded as an $(R,R)$-bimodule.

References

[a1]	Paul M. Cohn. Basic Algebra: Groups, Rings, and Fields, Springer (2003) ISBN 1852335874. Zbl 1003.00001