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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The gluing of maximal ideals--spectrum of a Noetherian ring--going up and going down in polynomial rings

Authors: Ada Maria de Souza Doering and Yves Lequain
Journal: Trans. Amer. Math. Soc. 260 (1980), 583-593
MSC: Primary 13E05
MathSciNet review: 574801
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Abstract: If $ {M_1},\,...\,,\,{M_s}$ 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 $ {M_1}\, \cap \,D\, = \,...\, = \,{M_s}\, \cap \,D$ can be constructed.

We use this gluing process to prove the following result: Given any finite ordered set $ \mathcal{B}$, there exists a reduced Noetherian ring B and an embedding $ \psi :\,\mathcal{B}\, \to \,Spec\,B $ such that $ \psi $ establishes a bijection between the maximal (respectively minimal) elements of $ \mathcal{B}$ and the maximal (respectively minimal) prime ideals of B and such that given any elements $ \beta '$, $ \beta ''$ of $ \mathcal{B}$, there exists a saturated chain of prime ideals of length r between $ \psi (\beta ')$ and $ \psi (\beta '')$ if and only if there exists a saturated chain of length r between $ \beta '$ and $ \beta ''$.

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: $ A\,\hookrightarrow \,B$ possesses the Going Up and the Going Down properties, $ A[X]\,\hookrightarrow \,B[X]$ is unibranched and $ A[X]\,\hookrightarrow \,B[X]$ possesses neither the Going Up nor the Going Down properties.

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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