Prime ideal

This article is about a basic definition in commutative algebra. View a complete list of basic definitions in commutative algebra

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

Symbol-free definition

An ideal in a commutative unital ring is termed a prime ideal if it is proper, and 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
Whenever the product of two ideals is contained in it, one of the ideals is contained in it
It is an ideal whose complement is a saturated subset (that is, is closed with respect to the operations of multiplication and factorization).
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 $I$ is proper in $R$ and it satisfies the following equivalent conditions:

Whenever $a,b\in R$ are such that $ab\in I$ then either $a\in I$ or $b\in I$
Whenever $J$ and $K$ are ideals such that $JK\subset I$ , then either $J\subset I$ or $K\subset I$
The complement of $I$ in $R$ is a multiplicatively closed and saturated subset i.e. $ab\in R\setminus I\iff a\in R\setminus I$ or $b\in R\setminus I$
The quotient ring $R/I$ is an integral domain

Relation with other properties

Stronger properties

Weaker properties

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.

Contraction-closedness

This property of ideals in commutative unital rings is contraction-closed: a contraction of an ideal with this property, also has this property
View other contraction-closed properties of ideals in commutative unital rings

If $f:R\to S$ is a homomorphism of commutative unital rings, and $I$ is a prime ideal of $S$ , then the contraction of $I$ to $R$ , denoted $I^{c}$ , (same as $f^{-1}(I)$ ) is a prime ideal in $R$ .

For full proof, refer: primeness is contraction-closed

Note that this implies the intermediate subring condition and the transfer condition on ideals.

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.