Zero-dimensional ring

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 space

View 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

Stronger properties

• Finite ring
• Field
• Finite-dimensional algebra over a field
• Artinian ring

Weaker properties

• Jacobson ring
• Cohen-Macaulay ring: For full proof, refer: Zero-dimensional implies Cohen-Macaulay
• Equidimensional ring