Primary implies prime radical

This article gives the statement (and possibly proof) of an implication between properties of ideals in commutative unital rings; viz., any ideal in a commutative unital ring satisfying the first property, also satisfies the second

Statement

Property-theoretic statement

The property of ideals of being a primary ideal is stronger than the property of being an ideal with prime radical.

Verbal statement

The radical of a primary ideal is prime.

Converse

The converse of the statement is not true. For full proof, refer: prime radical not implies primary

However, any ideal with maximal radical is primary; in other words, if the radical is a maximal ideal, the ideal we started with is primary. For full proof, refer: maximal radical implies primary

Proof

Suppose $R$ is a commutative unital ring and $Q$ is a primary ideal in $R$. Let $P$ be the radical of $Q$. We need to show that $P$ is prime in $R$.

Pick $a,b\in R$ such that $ab\in P$. Then, since $P$ is the radical of $Q$, there exists $n\in \mathbb {N}$ such that $(ab)^{n}\in Q$. Hence $a^{n}b^{n}\in Q$, so either $a^{n}\in Q$ or there exists $m$ such that $(b^{n})^{m}=b^{nm}\in Q$. In the first case, $a\in P$, and in the second case, $b\in P$.