Let $R$ be a commutative ring with identity. If $P$ is a prime ideal of $R$ that is minimal over some zerodivisor of $R$, then must $P$ consist only of zerodivisors? I suspect not but I can't figure out how to construct a counterexample.
(The full context of the situation I am in is this. Suppose that $P$ is a prime ideal of $R$ such that $PP^{-1} \neq P$. Then there is an element $a$ of $P$ and an element $b$ of $R-P$ such that $P = (aR :_R bR)$. In this case, $P$ is necessarily minimal over $a$. My question is, if $a$ is a zerodivisor, can I conclude that $P$ consists only of zerodivisors?)