Lesson plan
Catenary ring
In mathematics, a commutative ring R is catenary if for any pair of prime ideals p, q, any two strictly increasing chains :p = p 0 ⊂ p 1 ⊂ ... ⊂ p n = q of prime ideals are contained in maximal strictly increasing chains from p to q of the same (finite) length. In a geometric situation, in which the dimension of an algebraic variety attached to a prime ideal will decrease as the prime ideal becomes bigger, the length of such a chain n is usually the difference in dimensions. A ring is called universally catenary if all finitely generated algebras over it are catenary rings. The word 'catenary' is derived from the Latin word catena, which means "chain". There is the following chain of inclusions.
Article summary
In mathematics, a commutative ring R is catenary if for any pair of prime ideals p, q, any two strictly increasing chains :p = p 0 ⊂ p 1 ⊂ ... ⊂ p n = q of prime ideals are contained in maximal strictly increasing chains from p to q of the same (finite) length. In a geometric situation, in which the dimension of an algebraic variety attached to a prime ideal will decrease as the prime ideal becomes bigger, the length of such a chain n is usually the difference in dimensions. A ring is called universally catenary if all finitely generated algebras over it are catenary rings. The word 'catenary' is derived from the Latin word catena, which means "chain". There is the following chain of inclusions.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
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- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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