Zero-dimensional ring
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: {{{1}}}
View other properties of commutative unital rings determined by the spectrum
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