# When is the tensor product commutative?

I am working with the the tensor product $-\otimes_R -$ over some noncommutative ring $R$. Is the tensor product always commutative if $R$ is commutative? If so, can the tensor product be commutative if $R$ is noncommutative?

• Do you mean whether $M\otimes_RN\simeq N\otimes_RM$? – user26857 Jul 6 '15 at 16:59
• The thing is, if $R$ is not commutative, $M\otimes_R N$ makes sense only if $M$ is a right $R$-module and $N$ a left $R$-module. Hence the tensor product is not commutative, but worse, it is defined for only one way. – Roland Jul 6 '15 at 17:48
• Well, I suppose we could consider $M,\,N$ to be $R$-$R$- bimodules? – Sam Williams Jul 7 '15 at 14:05

• Actually, even if $R$ is commutative, then the tensor product is not always commutative! Indeed, we are talking about $R$-$R$-bimodules here, not about $R$-modules (= $R$-$R$-bimodules satisfying the identity $rm = mr$). In $M \otimes_R N$, you "use up" the right $R$-module structure on $M$ and the left $R$-module structure on $N$ for tensoring, whereas in $N \otimes_R M$ it is the "other two" structures that you are using. The results can easily be non-isomorphic. – darij grinberg Sep 13 '15 at 23:16