An ideal $Q$ in a commutative ring $R$ is a primary ideal if for all elements $x,y\in R$ , we have that if $xy\in Q$ , then either $x\in Q$ or $y^n\in Q$ for some $n\in\mb{N}$ .

This is clearly a generalization of the notion of a prime ideal, and (very) loosely mirrors the relationship in $\mb{Z}$ between prime numbers and prime powers.

It is clear that every prime ideal is primary.

Example. Let $Q=(25)$ in $R=\mb{Z}$ . Suppose that $xy\in Q$ but $x\notin Q$ . Then $25|xy$ , but 25 does not divide $x$ . Thus 5 must divide $y$ , and thus some power of $y$ (namely, $y^2$ ), must be in $Q$ .

The radical of a primary ideal is always a prime ideal. If $P$ is the radical of the primary ideal $Q$ , we say that $Q$ is $P$ -primary.