Talk:Bimodule

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 $R\otimes _{\mathbb {Z} }S^{\mathrm {op} }$ module and vice versa.

mistakes: categories of bimodules and left modules

I believe the statement that "$Bimod(R)=R-Mod$" for a commutative ring $R$ is wrong. These categories are not canonically equivalent: while any left $R$ -module can be made into a bimodule by taking the same action on the right, it is not true that any bimodule has the same action on left and right sides. So there is just a faithful monoidal functor $R-Mod\rightarrow Bimod(R)$ .

Also, in this case, the monoidal structure on $R-Mod$ is indeed symmetric in an obvious way, but the monoidal structure on $Bimod(R)$ isn't! Here it might make sense to mention the fact that $Bimod(R)$ is not equivalent to $R\otimes _{Z}R-Mod$ as monoidal categories (different tensor product).

Can someone please fix these two mistakes in the article?

132.76.50.6 (talk) 08:59, 6 October 2015 (UTC)Inna Entova

Weak 2-category

Where it says:

> This is in fact a 2-category, in a canonical way...

I think it should say *weak* 2-category, since 1-cell composition is not strict as noted above, and noted here. — Preceding unsigned comment added by Jesuslop (talk • contribs) 08:54, 16 September 2018 (UTC)

Are there any plans to fix this mistake? 2003:DC:3F31:4121:4E6:7E16:B528:D086 (talk) 09:40, 13 February 2020 (UTC)