Zero-dimensional ring
From Commalg
This article is about a standard (though not very rudimentary) definition in commutative algebra. The article text may, however, contain more than just the basic definition
View a complete list of semi-basic definitions on this wiki
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
This property of commutative unital rings is completely determined by the spectrum, viewed as an abstract topological space. The corresponding property of topological spaces is: T1 spaceView other properties of commutative unital rings determined by the spectrum
Any integral domain satisfying this property of commutative unital rings, must be a field
Definition
Symbol-free definition
A commutative unital ring is termed zero-dimensional if it satisfies the following equivalent conditions:
- It has Krull dimension zero
- Every prime ideal in it is maximal
- Any quotient ring of it that is an integral domain is also a field
- The spectrum of the ring is a T1 space i.e. all points in the spectrum are closed
Relation with other properties
Conjunction with other properties
- Zero-dimensional Noetherian ring: A zero-dimensional ring that is also a Noetherian ring.
Stronger properties
Weaker properties
- Jacobson ring: For proof of the implication, refer Zero-dimensional implies Jacobson and for proof of its strictness (i.e. the reverse implication being false) refer Jacobson not implies zero-dimensional
- Cohen-Macaulay ring (under the assumption that the ring is Noetherian): For full proof, refer: Zero-dimensional Noetherian implies Cohen-Macaulay
- Equidimensional ring
