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&times;Sop, where Sop is the opposite ring of S (with the multiplication turned around). Bimodule homomorphisms are the same as homomorphisms of left R&times;Sop 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×Zop 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)