primary ideal

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 $n\in \mathbb{N}$.

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

It is clear that every prime ideal is primary.

Example. Let $Q=(25)$ in $R=\mathbb{Z}$. 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.