Primary ideal

Let $R$ be a commutative ring. A primary ideal $Q\subsetneq R$ is an ideal such that whenever $ab\in Q$ (with $a,b\in R$), then either $a\in Q$ or $b^n\in Q$ for some integer $n\ge 1$.

Primary ideals interpolate between prime ideals and powers: the radical $\sqrt{Q}$ is always a prime ideal , and elements outside $\sqrt{Q}$ act as non-zerodivisors on $R/Q$ up to nilpotence .

Examples:

• In $\mathbb{Z}$, the ideal $(p^n)$ is primary for any prime $p$ and $n\ge 1$.
• In $k[x,y]$, the ideal $(x^2,y)$ is primary; its radical is $(x,y)$.
• The ideal $(xy)\subset k[x,y]$ is not primary: $x\cdot y\in (xy)$ but $x\notin (xy)$ and $y^n\notin (xy)$ for all $n\ge 1$.