Prime 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: integral domain | View other quotient-determined properties of ideals in commutative unital rings

Definition for commutative rings

Symbol-free definition

An ideal in a commutative unital ringis termed a prime ideal if it satisfies the following equivalent conditions:

• Whenever the product of two elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal.
• It is an ideal whose complement is a saturated subset (that is, is clsoed with respect to the operation of multiplication).
• The quotient ring by that ideal is an integral domain

Definition with symbols

An ideal $I$ in a commutative unital ring $R$ is termed a prime ideal if whenever $a,b\in R$ are such that $ab\in I$ then either $a\in I$ or $b\in I$.

Relation with other properties

Stronger properties

• Maximal ideal
• Minimal prime ideal

Weaker properties

• Irreducible ideal
• Primary ideal
• Radical ideal

Metaproperties

Intersection-closedness

This property of ideals in commutative unital rings is not closed under taking arbitrary intersections; in other words, an arbitrary intersection of ideals with this property need not have this property

An intersection of prime ideals need not be prime. In fact, an ideal is an intersection of prime ideals iff it is a radical ideal.

Intermediate subring condition

This property of ideals satisfies the intermediate subring condition for ideals: if an ideal has this property in the whole ring, it also has this property in any intermediate subring
View other properties of ideals satisfying the intermediate subring condition

If an ideal is prime in the whole ring, it is also prime in any intermediate subring. This is related to the fact that any subring of an integral domain is an integral domain.

Transfer condition

This property of ideals satisfies the transfer condition for ideals: if an ideal satisfies the property in the ring, its intersection with any subring satisfies the property inside that subring

If $I$ is a prime ideal in $R$, and $S$ is any subring of $R$, then $I\cap S$ is a prime ideal in $S$. Note that this implies the intermediate ring condition as well.

Effect of property operators

The intersection-closure

Applying the intersection-closure to this property gives: radical ideal

An ideal in a commutative unital ring is expressible as an intersection of prime ideals iff it is a radical ideal.