On commutative reduced local rings

Question

If $R$ is a reduced local commutative ring with maximal ideal $\mathfrak m$ which is also an associated prime ideal ($\mathfrak m=\operatorname{Ann}_R(a)$ for some $a\in R$). How can I prove that $R$ is a field?

Answer 1

If we show that $0\neq a$ is in all prime ideals, you would have the required contradiction, since intersection of all prime ideals is the nil radical. If $a\not\in I$, then $a$ is unit and then $Ia=0$ implies $I=0$, proving what you need. If $a$ is not in a prime ideal $P\neq I$, then, since $0=Ia\subset P$, we get $I\subset P$. Hope rest is clear.