Talk:Bimodule

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Mathematics rating:	Start Class	Mid Priority	Field: Algebra
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I believe this paragraph is false: "An R-S bimodule is actually the same thing as a left module over the ring R×S^op, where S^op is the opposite ring of S (with the multiplication turned around). Bimodule homomorphisms are the same as homomorphisms of left R×S^op modules. Using these facts, many definitions and statements about modules can be immediately translated into definitions and statements about bimodules. For example, the category of all R-S bimodules is abelian, and the standard isomorphism theorems are valid for bimodules."

Consider Z as the obvious Z-Z bimodule. Then the Z×Z^op module structure on Z should be defined as (a, b)c = acb. But, for example, ((0, 1) + (1, 1))1 = (1, 2)1 = 2, while (0, 1)1 + (1, 1)1 = 0 + 1, so distributivity fails.

It is true, however, that an R-S bimodule can be regarded as a left module and vice versa.