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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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Prime ideals in differential operator rings. Catenarity
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by K. A. Brown, K. R. Goodearl and T. H. Lenagan PDF
Trans. Amer. Math. Soc. 317 (1990), 749-772 Request permission

Abstract:

Let $R$ be a commutative algebra over the commutative ring $k$, and let $\Delta = \{ {\delta _1}, \ldots ,{\delta _n}\}$ be a finite set of commuting $k$-linear derivations from $R$ to $R$. Let $T = R[{\theta _1}, \ldots ,{\theta _n};{\delta _1}, \ldots ,{\delta _n}]$ be the corresponding ring of differential operators. We define and study an isomorphism of left $R$-modules between $T$ and its associated graded ring $R[{x_1}, \ldots ,{x_n}]$, a polynomial ring over $R$. This isomorphism is used to study the prime ideals of $T$, with emphasis on the question of catenarity. We prove that $T$ is catenary when $R$ is a commutative noetherian universally catenary $k$-algebra and one of the following cases occurs: (A) $k$ is a field of characteristic zero and $\Delta$ acts locally finitely; (B) $k$ is a field of positive characteristic; (C) $k$ is the ring of integers, $R$ is affine over $k$, and $\Delta$ acts locally finitely.
References
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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 317 (1990), 749-772
  • MSC: Primary 16A05; Secondary 16A66
  • DOI: https://doi.org/10.1090/S0002-9947-1990-0946215-3
  • MathSciNet review: 946215