Zero-dimensional ring: Difference between revisions
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* Every [[prime ideal]] in it is [[maximal ideal|maximal]] | * Every [[prime ideal]] in it is [[maximal ideal|maximal]] | ||
* Any [[quotient ring]] of it that is an integral domain is also a field | * Any [[quotient ring]] of it that is an integral domain is also a field | ||
* The [[spectrum of a commutative unital ring|spectrum]] of the ring is a [[tps:T1 space|T1 space]] i.e. all points in the spectrum are closed | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 01:32, 10 January 2008
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