Tensor product of modules over non commutative rings

You can do the same thing for a noncommutative ring; it's just not as useful and so is not a standard definition. Notice that these relations imply that $$rs(x\otimes y)=r(sx\otimes y)=sx\otimes ry=s(x\otimes ry)=sr(x\otimes y)$$ for any $r,s\in R$ and any $x\in M$, $y\in N$. So $R$ will act "commutatively" on the tensor product "$M\otimes N$" defined in this way: the action will factor through the quotient $R/[R,R]$ by the commutator ideal. So constructing tensor products in this way loses all information about the noncommutativity of $R$ (and of its action on the modules $M$ and $N$). This is rarely useful when thinking about noncommutative rings.

(Indeed, even if you do want to talk about this construction for a noncommutative ring, you don't need to, since you can define it just using the tensor product of modules over a commutative ring. For the tensor product of $M$ and $N$ defined in this way is naturally isomorphic to the tensor product $M/[R,R]M\otimes_{R/[R,R]} N/[R,R]N$ of $R/[R,R]$-modules, with its natural $R$-module structure.)