## Primary Ideals, Prime Ideals, and Radical Ideals

## Problem 247

Let $R$ be a commutative ring with unity. A proper ideal $I$ of $R$ is called **primary** if whenever $ab \in I$ for $a, b\in R$, then either $a\in I$ or $b^n\in I$ for some positive integer $n$.

**(a)** Prove that a prime ideal $P$ of $R$ is primary.

**(b)** If $P$ is a prime ideal and $a^n\in P$ for some $a\in R$ and a positive integer $n$, then show that $a\in P$.

**(c)** If $P$ is a prime ideal, prove that $\sqrt{P}=P$.

**(d)** If $Q$ is a primary ideal, prove that the radical ideal $\sqrt{Q}$ is a prime ideal.