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 is termed primary if it satisfies the following equivalent conditions:

• 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
• There is exactly one associated prime to the ideal, i.e. exactly one associated prime to the quotient ring

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