Primary ideal

Definition for commutative rings

Symbol-free definition

An ideal in a commutative unital ring (or in any commutative ring) is termed primary if it satisfies the condition that whenever the product of two elements of the ring lies inside the ideal, either the first element lies inside the ideal or a suitable power of the second element lies inside the ideal.

Definition with symbols

An ideal $I$ in a commutative ring $R$ is termed primary if for any $p,q$ in $R$ such that $pq$ is in $I$, either $p$ is in $I$, or there exists a positive integer $n$ such that $q^{n}$ lies in $I$.

Definition for non-commutative rings

The same definition applies verbatim for noncommutative rings. See Primary ideal (noncommutative rings) for more details.