Bimodule - Examples

Examples

• For positive integers n and m, the set M_n,m(R) of n × m matrices of real numbers is an R-S bimodule, where R is the ring M_n(R) of n × n matrices, and S is the ring M_m(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 M_n,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.
• If R is a ring, then R itself is an R-bimodule, and so is Rn (the n-fold direct product of R).
• Any two-sided ideal of a ring R is an R-bimodule.
• Any module over a commutative ring R is automatically 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. (However, not all R-bimodules arise this way.)
• If M is a left R-module, then M is an R-Z bimodule, where Z is the ring of integers. Similarly, right R-modules may be interpreted as Z-R bimodules, and indeed an abelian group may be treated as a Z-Z bimodule.
• If R is a subring of S, then S is an R-bimodule. It is also an R-S and an S-R bimodule.