Primary ideal

This article defines a property of an ideal in a commutative unital ring

This property of an ideal in a ring is equivalent to the property of the quotient ring being a/an: primary ring | View other quotient-determined properties of ideals in commutative unital rings

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.