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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The gluing of maximal ideals—spectrum of a Noetherian ring—going up and going down in polynomial rings
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by Ada Maria de Souza Doering and Yves Lequain PDF
Trans. Amer. Math. Soc. 260 (1980), 583-593 Request permission

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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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 260 (1980), 583-593
  • MSC: Primary 13E05
  • DOI: https://doi.org/10.1090/S0002-9947-1980-0574801-5
  • MathSciNet review: 574801