Integral domain

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 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

Definition

Symbol-free definition

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

• It is a nonzero ring, and is cancellative
• The zero ideal is a prime ideal
• It is a nonzero ring, and the product of nonzero elements in nonzero
• It is a nonzero ring, and the product of nonzero ideals is nonzero
• It is a nonzero ring, and the set of nonzero elements is a saturated subset

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.

Closure under taking quotients

This property of commutative unital rings is not quotient-closed: in other words, a quotient of a commutative unital ring with this property, need not have this property

A quotient of an integral domain by an ideal need not be an integral domain; to see this, note that the quotient by an ideal is an integral domain iff the ideal is a prime ideal. Thus, the quotient of $\mathbb {Z}$ by the non-prime ideal $6\mathbb {Z}$ is not an integral domain.

Effect of property operators

The quotient-closure

Applying the quotient-closure to this property gives: commutative unital ring

Any commutative unital ring can be expressed as a quotient of an integral domain by an ideal. In fact, any commutative unital ring can be expressed as a quotient of a polynomial ring over integers (a free commutative unital ring) by a suitable ideal.

External links

Definition links

• Wikipedia page
• Mathworld page
• Planetmath page