Bimodule

Remarks

An $(R, R)$-bimodule is also called an $R$-bimodule. Example: for a field $K$, the space $M_{m \times n}(K)$ of $m \times n$ matrices is an $(M_m(K), M_n(K))$-bimodule under matrix multiplication. Clauses 1 and 2 are the usual rules of matrix arithmetic, and clause 3 is associativity $(AX)B = A(XB)$. Every ring $R$ is an $R$-bimodule, and its sub-bimodules are its two-sided ideals. Non-example: let $R$ be a noncommutative ring and give $R$ the "right action" $m \cdot s = sm$. This fails clause 2, since $(m \cdot s) \cdot t = tsm$ while $m \cdot (st) = stm$, and $ts \ne st$ for some $s, t$. For a commutative ring $K$, every $K$-module is a $K$-bimodule with $mk = km$.