Definition Let R and S be unital rings. An R-S-bimodule is an Abelian group M, where elements of M may be multiplied on the left by elements of R, and may also be multiplied on the right by elements of S, and where the following properties are satisfied:

(i) M is a left R-module;

(ii) M is a right S-module;

(iii) (rx)s = r(xs) for all x ∈ M, r ∈ R and s ∈ S.

Example Let K be a field, let m and n be positive integers, and let Mm,n(K)  denote the set of m × n matrices with coefficients in the field K. Then Mm,n(K) is an Abelian group with respect to the operation of matrix addition.

The elements of M_m,n(K) may be multiplied on the left by elements of the ring M_m(K) of m × m matrices with coefficients in K; they may also be multiplied on the right by elements of the ring M_n(K) of n × n matrices with coefficients in K; these multiplication operations are the usual ones resulting from matrix multiplication. Moreover (AX)B = A(XB) for all X ∈ M_m,n(K), A ∈ M_m(K) and B ∈ M_n(K). Thus M_m,n(K) is an M_m(K)-M_n(K)-bimodule.

If R is a unital commutative ring then any R-module M may be regarded as an R-R-bimodule, where (rx)s = r(xs) = (rs)x for all x ∈ M and r, s ∈ R.

Definition Let R and S be unital rings, and let M and N be R-S-bimodules.