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In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in the sense that many of the relationships between left and right modules become simpler when they are expressed in terms of bimodules. If R and S are two rings, then an R-S-bimodule is an abelian group such that: M is a left R-module and a right S-module. For all r in R, s in S and m in M: An R-R-bimodule is also known as an R-bimodule. For positive integers n and m, the set Mn,m(R) of n × m matrices of real numbers is an R-S-bimodule, where R is the ring Mn(R) of n × n matrices, and S is the ring Mm(R) of m × m matrices. Addition and multiplication are carried out using the usual rules of matrix addition and matrix multiplication; the heights and widths of the matrices have been chosen so that multiplication is defined. Note that Mn,m(R) itself is not a ring (unless n = m), because multiplying an n × m matrix by another n × m matrix is not defined. The crucial bimodule property, that (r.x).s = r.(x.s), is the statement that multiplication of matrices is associative (which, in the case of a matrix ring, corresponds to associativity). Any algebra A over a ring R has the natural structure of an R-bimodule, with left and right multiplication defined by and respectively, where is the canonical embedding of R into A. If R is a ring, then R itself can be considered to be an R-R-bimodule by taking the left and right actions to be multiplication—the actions commute by associativity. This can be extended to Rn (the n-fold direct product of R). Any two-sided ideal of a ring R is an R-R-bimodule, with the ring multiplication both as the left and as the right multiplication. Any module over a commutative ring R has the natural structure of a bimodule. For example, if M is a left module, we can define multiplication on the right to be the same as multiplication on the left.

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