Tensor products and linear maps

Question

Let $R$ be a commutative ring (with unity) and let $M,M',N,N'$ be $R$-modules. I know that there is a standard linear map $$\varphi:\,Hom_R(M,M')\oplus Hom_R(N,N')\longrightarrow Hom_R(M\otimes_R N,\, M'\otimes_R N')$$ sending $(\alpha,\beta)$ to $\alpha\otimes\beta$ and this last map acts as $m\otimes n\mapsto \alpha(m)\otimes \beta(n)$ on elementary tensor products. I know that $\varphi$ is not injective in general, but I cannot find an example in which the map $\varphi$ is not surjective. Can you help me find one?

Answer 1

Assume $M,M',N,N'$ are free $R$-modules of rank $m,m',n,n'$ respectively. Then your source module is free of rank $mm'+nn'$, while your target module is free of rank $mnm'n'$. If $mm'+nn'< mnm'n'$, your map cannot be surjective. For a concrete example, take $m=m'=n=n'=d$, where $d\geq 2$. Then $mm'+nn'=2d^2<mnm'n'=d^4$.