Consider a noncommutative ring with unity $R$, three left $R$-modules $M,N,P$ and a map $f\colon\;M\times N\to P$ such that:
$ f(m+m',n)=f(m,n)+f(m',n)\\ f(m,n+n')=f(m,n)+f(m,n')\\ f(rm,n)=rf(m,n)=f(m,rn) $
A map like this does exist: the null one. What about others? Can exist a non-null map satisfying those axioms? From the third point, given two scalars $r,s\in R$, it is evident that $(rs-sr)f(m,n)=0$ for all $m,n$ and this could be a problem given that $R$ is non-commutative.