In commutative algebra, is it true that the tensor product of two faithful modules is a faithful module?
I have written for myself a proof for the case of finitely generated modules over reduced rings: Suppose $M,N$ are finite $A$-modules, with $A$ reduced. Then
$\mathrm{V(Ann}(M\otimes_A N))=\mathrm{Supp}(M\otimes_A N)=\mathrm{Supp(M)}\cap\mathrm{Supp(N)}=\mathrm{V}(\mathrm{Ann}(M))\cap\mathrm{V(Ann}(N))=\mathrm{V}(0)\cap\mathrm{V}(0)=\mathrm{Spec}(A),$
and hence $\sqrt{\mathrm{Ann}(M\otimes_A N)}=\sqrt{(0)}=(0),$ hence $\mathrm{Ann}(M\otimes_A N)=0$.