One-dimensional ring

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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

A commutative unital ring is termed one-dimensional if it satisfies the following equivalent conditions:

  • Its Krull dimension is one
  • Every prime ideal in it is either a minimal prime ideal or a maximal ideal, and not every prime ideal is both (i.e. there exists at least one minimal prime ideal that is not maximal, or at least one maximal ideal that is not a minimal prime)

Relation with other properties

Conjunction with other properties