When defining tensor products $M\otimes_R N$ over a commutative ring $R$ one can use a universal property with respect to bilinear maps $M\times N\rightarrow P$.
On the other hand, in the general case, for noncommutative rings one has to use balanced maps $M \times N \rightarrow Z$ instead of bilinear. Of course, in the first case $P$ is an $R$-module while in the second case $Z$ is just an abelian group.
Remember $f$ is biliniar if $f(mr, n)=rf(m, n)=f(m, nr)$ while $f$ is balanced if and only if $f(mr,n)=f(m, rn)$.
I have the following two natural questions:
- Why the two definitions coincide?
- Is there an example of a balanced map $M\times N \rightarrow P$ which is not bilinear? I cannot construct one by myself. Here I assume R commutative so I can speak about bilinear maps.