One-dimensional domain

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
View other properties of integral domains | 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

An integral domain is termed a one-dimensional domain if it satisfies the following equivalent conditions:

• Every nonzero prime ideal in it is maximal
• It has Krull dimension at most one (note that the Krull dimension is zero iff it is a field)

Relation with other properties

Stronger properties

• Field
• Polynomial ring over a field
• Ring of integers in a number field
• Principal ideal domain
• Dedekind domain

Weaker properties

• Finite-dimensional domain
• Finite-dimensional ring