It is known that if $R$ is a commutative ring and $M\otimes_R N=0$ for two finitely generated $R$-modules $M$ and $N$, then $Ann(M)+Ann(N)=R$, where $Ann()$ stands for the annihilator. The proof that I know uses an argument involving determinants of matrices with entries in $R$ and is therefore dependent to the commutativity of $R$.
I want to know what happens if I drop the commutativity hypothesis? Clearly, the annihilators $Ann(M)$ and $Ann(N)$ are two-sided ideals and one may still form the sum $Ann(M)+Ann(N)$. Is it true that $Ann(M)+Ann(N)=R$ in this case?