I am confused of the definition of tensor product of modules over a non commutative ring.
First let $R$ be a commutative ring and let $M$ and $N$ be two $R$ modules. Let $F_{R}(M\times N)$ be the free $R$ module over the set $M\times N$ and let $K$ be the submodule generated by elements of the form
$(x,y_1+y_2)-(x,y_1)-(x,y_2)$,
$(x_1+x_2,y)-(x_1,y)-(x_2,y)$,
$(rx,y)-r(x,y)$,
$(x,ry)-r(x,y)$. Then the tensor product is $F_{R}(M\times N)/K$. My doubt is why cant we do the same if R is a non commutative ring? Where is the commutativity of $R$ is used above? Please help me.
Best Answer
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.)