## Every Prime Ideal is Maximal if $a^n=a$ for any Element $a$ in the Commutative Ring

## Problem 530

Let $R$ be a commutative ring with identity $1\neq 0$. Suppose that for each element $a\in R$, there exists an integer $n > 1$ depending on $a$.

Then prove that every prime ideal is a maximal ideal.

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