Let $M$ and $N$ be modules over a ring $R$. Generally, the tensor product $M\otimes N$ is defined to be an abelian group with a balanced map $j:M\times N\to M\otimes N$ such that for any abelian group $G$ with a balanced map $i:M\times N\to G$ there is a unique homomorphism $\phi:M\otimes N\to G$ such that $\phi\circ j=i$.
My question is why not to define the tensor product as following:
Suppose $M$ and $N$ are left $R$-modules, the tensor product $M\otimes N$ is defined to be a left $R$-module with a bilinear map $j:M\times N\to M\otimes N$ such that for any $R$-module $G$ with a bilinear map $i:M\times N\to G$ there is a unique homomorphism $\phi:M\otimes N\to G$ such that $\phi\circ j=i$.
One reason is this leads to modules that behave poorly with respect to commutators. For example, let $m\in M$, $n\in N$, and suppose $r,s\in R$ don't commute. Then $$rm\otimes sn=rs(m \otimes n)=sr(m \otimes n)$$ So any commutator $rs-sr$ annihilates the whole tensor product.