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universally accessible key - translation to russian
![ii-V<sup>7</sup>-I progression]] in C [[File:Ii-V-I turnaround in C.mid]]](https://commons.wikimedia.org/wiki/Special:FilePath/Ii-V-I turnaround in C.png?width=200 "ii-V<sup>7</sup>-I progression]] in C [[File:Ii-V-I turnaround in C.mid]]")
Definition
Wikipedia
In mathematics, a commutative ring R is catenary if for any pair of prime ideals p, q, any two strictly increasing chains
- p = p0 ⊂ p1 ⊂ ... ⊂ pn = 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.
- Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings
