Principal ideal ring

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

Definition

Symbol-free definition

A commutative unital ring is termed a principal ideal ring if every ideal in it is principal, that is, if every ideal is generated by a single element.

Definition with symbols

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Relation with other properties

Conjunction with other properties

• Principal ideal domain is a principal ideal ring which is also an integral domain

Weaker properties

• Bezout ring
• Noetherian ring

Metaproperties

Closure under taking quotient rings

This property of commutative unital rings is quotient-closed: the quotient ring of any ring with this property, by any ideal in it, also has this property

View other quotient-closed properties of commutative unital rings