$M\otimes_A N = M\otimes_{A/a} N $? Let $A$ be a commutative ring and $a$ an ideal. Let $M$ and $N$ be $A$-modules. 
Now suppose $a\subset Ann(M)\cap Ann(N)$. I'm hoping that $M\otimes_A N = M\otimes_{A/a} N$. Is it true?

Yes, it is true. For any $A/\mathfrak a$-module $P$, $A$-bilinear maps $M\times N \to P$ are clearly the same thing as $A/\mathfrak a$-bilinear maps $M\times N\to P$, since $M$ and $N$ are killed by $\mathfrak a$. 
By the universal property of the tensor product and by the Yoneda lemma, it follows that there is a canonical isomorphism $M\otimes_A N \simeq M \otimes_{A/\mathfrak a} N$.
