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

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.

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.

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.