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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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Intermediate normalizing extensions
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by A. G. Heinicke and J. C. Robson PDF
Trans. Amer. Math. Soc. 282 (1984), 645-667 Request permission

Abstract:

Relationships between the prime ideals of a ring $R$ and of a normalizing extension $S$ have been studied by several authors recently. In this work, most of these known results are extended to give relationships between the prime ideals of $R$ and of $T$ where $T$ is a ring with $R \subset T \subset S$, and $S$ is a normalizing extension of $R$: such rings $T$ are called intermediate normalizing extensions of $R$. One result ("Cutting Down") asserts that for any prime ideal $J$ of $T$, $J \cap R$ is the intersection of a finite set of prime ideals ${P_i}$ of $R$, uniquely defined by $J$, whose corresponding factor rings $R/{P_i}$ are mutually isomorphic. The minimal members of the family of ${P_i}$’s are the primes of $R$ minimal over $J \cap R$, and an "incomparability" theorem is proved which shows that no two comparable primes of $T$ can have their intersections with $R$ share a common minimal prime. Other results include versions of the "lying over" and "going up" theorems, proofs that chain conditions such as right Goldie or right Noetherian pass between $T/J$ and each of the rings $R/{P_i}$, and a demonstration that the "additivity principle" holds.
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Additional Information
  • © Copyright 1984 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 282 (1984), 645-667
  • MSC: Primary 16A56; Secondary 16A26, 16A34, 16A55, 16A66
  • DOI: https://doi.org/10.1090/S0002-9947-1984-0732112-2
  • MathSciNet review: 732112