The gluing of maximal idealsspectrum of a Noetherian ringgoing up and going down in polynomial rings
Authors:
Ada Maria de Souza Doering and Yves Lequain
Journal:
Trans. Amer. Math. Soc. 260 (1980), 583593
MSC:
Primary 13E05
MathSciNet review:
574801
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Abstract: If are maximal ideals of a ring R that have isomorphic residue fields, then they can be ``glued'' in the sense that a subring D of R with R is integral over D and can be constructed. We use this gluing process to prove the following result: Given any finite ordered set , there exists a reduced Noetherian ring B and an embedding such that establishes a bijection between the maximal (respectively minimal) elements of and the maximal (respectively minimal) prime ideals of B and such that given any elements , of , there exists a saturated chain of prime ideals of length r between and if and only if there exists a saturated chain of length r between and . We also use the gluing process to construct a Noetherian domain A with quotient field L and a Noetherian domain B between A and L such that: possesses the Going Up and the Going Down properties, is unibranched and possesses neither the Going Up nor the Going Down properties.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198005748015
PII:
S 00029947(1980)05748015
Keywords:
Gluing of maximal ideals,
integral extension,
altitude formula,
finite ordered set,
spectrum of a Noetherian ring,
Going Up,
Going Down,
unibranchness,
polynomial ring
Article copyright:
© Copyright 1980
American Mathematical Society
