Catenary ring: Difference between revisions
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==Definition== | ==Definition== | ||
A [[commutative unital ring]] is said to be '''catenary''' if it satisfies the following condition: | A [[commutative unital ring]] is said to be '''catenary''' if it is [[Noetherian ring|Noetherian]] satisfies the following condition: | ||
If <math>P < P_1 < P_2 < Q</math> is a strictly ascending chain of [[prime ideal]]s, and <math>P'</math> is a prime ideal between <math>P</math> and <math>Q</math>, then there is either a prime ideal between <math>P</math> and <math>P'</math> or a prime ideal between <math>P'</math> and <math>Q</math>. | If <math>P < P_1 < P_2 < Q</math> is a strictly ascending chain of [[prime ideal]]s, and <math>P'</math> is a prime ideal between <math>P</math> and <math>Q</math>, then there is either a prime ideal between <math>P</math> and <math>P'</math> or a prime ideal between <math>P'</math> and <math>Q</math>. | ||
Revision as of 16:21, 30 June 2007
This article defines a property of commutative rings
Definition
A commutative unital ring is said to be catenary if it is Noetherian satisfies the following condition:
If is a strictly ascending chain of prime ideals, and is a prime ideal between and , then there is either a prime ideal between and or a prime ideal between and .
