Primary ideal

This article defines a property of an ideal in a commutative unital ring |View other properties of ideals in commutative unital rings

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

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 $I$.

Relation with other properties

Stronger properties

• Maximal ideal
• Prime ideal
• Ideal with maximal radical
• Irreducible ideal if the ring is Noetherian: For full proof, refer: Irreducible implies primary (Noetherian)

Weaker properties

• Ideal with prime radical

Incomparable properties

• Radical ideal