I've been trying to solve the following problem.
Let $A$ be a commutative ring with identity and $a \in A$ a non-nilpotent element, i.e., $a^m \neq 0$ for all $m \in \mathbb{Z}^+$. Prove there exists a prime ideal that fails to contain $a$.
I've been trying to somehow define a partial order on the prime ideals of $A$ to apply Zorn's lemma and proves the existence of such a prime ideal, but I am not sure this approach would work. Any ideas?