Integral domain

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

The property of being an ideal for which the quotient ring has this property is: prime ideal

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

Definition

Symbol-free definition

A commutative unital ring is termed an integral domain if it satisfies the following equivalent conditions:

• It is cancellative
• The zero ideal is a prime ideal
• The product of nonzero elements in nonzero

Definition with symbols

A commutative unital ring $R$ is termed an integral domain if $R$ satisfies the following equivalent conditions:

• Whenever $ab=ac$ and $a$ is not zero, $b=c$
• The ideal $0$ is a prime ideal
• Whenever $ab=0$, either $a=0$ or $b=0$

Relation with other properties

Stronger properties

• Field

Particular kinds of integral domains

Refer Category: Properties of integral domains

Weaker properties

• Reduced ring

Metaproperties

Closure under taking the polynomial ring

This property of commutative unital rings is polynomial-closed: it is closed under the operation of taking the polynomial ring. In other words, if $R$ is a commutative unital ring satisfying the property, so is $R[x]$

View other polynomial-closed properties of commutative unital rings

The polynomial ring over an integral domain is again an integral domain.

Closure under taking subrings

Any subring of a commutative unital ring with this property, also has this property
View other subring-closed properties of commutative unital rings

Any subring of an integral domain is an integral domain. In fact, a commutative unital ring is an integral domain iff it occursas as a subring of a field.