Let $I$ be a finitely generated ideal of a commutative ring $R$. Assume every element of $I$ is a zero divisor. Does then exist a $x \neq 0$ in $R$ with $xI=0$?
This is true if $0$ is a decomposable ideal, for example if $R$ is noetherian. I wonder if we actually need this. Doesn't it sound plausible? The problem is that we cannot just multiply the elements which kill the generators of $I$, the product can vanish.