Primary implies prime radical

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.

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.

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 $$a^nb^n \in Q$$, 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, $$b \in P$$.