Tensor product in a noncommutative division ring

Question

I have some problem understanding this proof by Qiaochu Yuan

During the proof of the double centralizer theorem, he wrote the tensor product of two module in this way.
Given $T\subset$ End$(A)$ and $T'$ the centralizer, i.e. $T'=$End$_T(A)$ then let $M$ be a simple left module of $T$ and $N=$hom$_T(M,A)$. Denote with $D=$End$(M)$ the associated division ring (noncommutative) then $M$ is a left $D$-module and $N$ is a right $D$-module.

He wrote at some point $M\otimes_D N$, but the general definition for the tensor product over a noncomm. ring S is for a right $V$ $S$-module and a left $W$ $S$-module.

My questions are:
(1) We can define the tensor product as in that proof?
(2) Is $M\otimes_D N$ still a left $T$-module whit the action $t(m\otimes n)=tm\otimes n$?

Answer 1

I mixed up my lefts and rights here, apologies. You can either rewrite the tensor product in the other order as $N_i \otimes M_i$ or replace $D_i$ with $D_i^{op}$ everywhere, as dsh says in the comments. I'll fix the post.