I have a question about next result.
Let R a ring $A'$ submodule of $A_R$, $B'$ submodule of $_RB$. Then $A/A'\otimes_R B/B'\cong (A\otimes_R B)/C$ where C is the subgroup of $A\otimes_RB$ generated by all elements $a'\otimes b$ and $a\otimes b'$ with $a\in A$, $b\in B$, $a'\in A'$ and $b'\in B'$
Partial: Let $\alpha: A\times B\to A/A'\otimes B/B'$ such that $(a,b)\mapsto a+A'\otimes b+B'$ this verify that $\alpha(a+x,b)=\alpha(a,b)+\alpha(x,b)\\\alpha(a,b+y)=\alpha(a,b)+\alpha(a,y)\\\alpha(ar,b)=\alpha(a,rb) $
Then there exist an unique $f:A\otimes B\to A/A'\otimes B/B'$ such that $f\iota=\alpha$
observe that $C\subseteq \ker(f)$ then there exist $g: (A\otimes B)/C\to A/A'\otimes B/B'$ such that $g(a\otimes b+C)=f(a\otimes b)$
I don't see that f is an surjective and that $\ker f\subseteq C$
Please help