Primary ideal

A proper ideal $I$ of a commutative ring is primary when $ab\in I$ and $a\notin I$ imply $b^n\in I$ for some positive integer $n$. Equivalently, every zero divisor of the nonzero quotient ring $R/I$ is nilpotent. Its radical of an ideal is a prime ideal.

For example, $(p^r)$ is primary in $\mathbb{Z}$ when $p$ is prime and $r\ge 1$. In contrast, $(6)$ is not primary: $2\cdot 3\in (6)$, but $2\notin (6)$ and no power of three lies in $(6)$. Requiring the ideal to be proper excludes the unit ideal, for which the elementwise implication would otherwise hold vacuously.