# notation questions $_AP$

I have $A$ and $B$ two rings and $P\in {_A\text{Mod}_B}$. What does $P_A$ mean and what does $_BP$ mean?

Also, what does $_B\text{End}(P)^{op}$ mean?

In the context of a Morita context $(B,A,P,Q,\mu,\tau)$. Then we have the following property:

$\mu$ is surjective

iff

$_BP$ is a generator, $A\cong {_B\text{End}(P)^{op}}$ and $Q\cong{_B\text{Hom}(P,B)}$

## 1 Answer

I would interpret $_AMod_B$ as the category of $A,B$ bimodules.

$P_A$ would mean a right $A$ module, and $_BP$ would mean a left $B$ module.

The last expression, $_B\mathrm{End}(P)^{op}$ is hard to interpret because it is unclear what $P$ is, and unclear what endomorphisms are being talked about, and unclear if it is referring to a module structure on the opposite ring, or the opposite ring of $B$ linear endomorphisms.

One simply cannot tell unless you provide more context.

From the context, it looks like $_B\mathrm{End}(P)$ is supposed to mean "the $B$ linear homomorphisms from $P$ to $P$." I think a rather clearer and more conventional way to write this is either as $\mathrm{End}_B(P)$ or better yet $\mathrm {End}(_BP)$. Writing the $B$ on the far left makes it look like notation for module structure (which is already being used) so it is a poor choice to put it there.

So in all, $\mathrm{End}(_BP)^{op}$ is the opposite ring of the ring of $B$ linear transformations on $P$.

• sorry, I added some context – tomak Dec 19 '17 at 17:28
• @tomak Great, I think that clears things up. – rschwieb Dec 19 '17 at 18:33