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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We provide a new description of the complex computing the Hochschild homology of an -unitary -algebra as a derived tensor product such that: (1) there is a canonical morphism from it to the complex computing the cyclic homology of that was introduced by Ko ...
2023
The aim of this article is to provide explicit formulas for the cup product on the Hochschild cohomology of any nonnegatively graded connected algebra and for the cap products on the Hochschild homology of with coefficients in any graded bimodule at the le ...
Let k be a field of characteristic /=2 and let W(k) be the Witt ring of k and L a finite extension of k. If L/k is a Galois extension, then the image of rL/k is contained in W(L)Gal(L/k) where rL/k:W(k)→W(L) is the canonical ring homomorphism. Rosenberg an ...