On Oct 15, 3:00 pm, Ralf Hemmecke <r...@hemmecke.de> wrote:
> Could you elaborate. I would like to understand why the statement that
> "every module is a bimodule" is not acceptable.
Because people tend to think "module = representation" and this is not
true for bimodules, or at least not true in the sense one would
expect.
In the context of representation theory, the term "modules" always
refer to one-sided modules but never to bimodules (cf. for instance
Pierce "Associative Algebras", ch. 2,). Intuitively one wants to think
of modules as representations of your ring (or algebra), and bimodules
are just not the right kind of object to do that. Even in the few
theories where A-A-bimodules are studied, like Hochschild (co)
homology, they are often considered as left modules over the
enveloping algebra A^e = A \otimes A^{op}.
For short, I would expect "Modules(R)" to be a sort of category of
representations of R, but if that category returns bimodules I am not
getting the representations of R, but the ones of its enveloping ring.
I might be wrong here, but I cannot think of a single math book where
the term "module" is understood as "bimodule".
Ideals are a different business. Their purpose is to provide
reasonable things to quotient out by (not representations), or if you
prefer categorical language a nice family of kernels. And kernels of
ring morphisms are always twosided ideals.
Is there any real reason to use "Modules(R)" as "Bimodules(R,R)"
beyond the comfort of not changing the names? If not, I'd strongly
suggest forgetting the name "Modules" for noncommutative rings or
default it to left modules.
Cheers
J
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