# One-dimensional ring

From Commalg

Revision as of 20:18, 5 May 2008 by Vipul (talk | contribs) (New page: {{curing property}} ==Definition== A commutative unital ring is termed '''one-dimensional''' if it satisfies the following equivalent conditions: * Its Krull dimension is one * ...)

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 ringsVIEW 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

- One-dimensional domain: This is a one-dimensional ring where there is a unique minimal prime ideal: the zero ideal
- One-dimensional local ring: This is a one-dimensional ring with a unique maximal ideal
- One-dimensional Noetherian ring: A one-dimensional ring that is also Noetherian
- One-dimensional reduced ring: This is a one-dimensional ring where the intersection of all minimal primes is zero