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 ring (or in any commutative ring) is 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 $b\in I$.

Definition for non-commutative rings

The definition has many different forms for noncommutative rings:

• Prime ideal (noncommutative rings)
• Completely prime ideal