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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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Krull dimension of differential operator rings. III. Noncommutative coefficients
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by K. R. Goodearl and T. H. Lenagan PDF
Trans. Amer. Math. Soc. 275 (1983), 833-859 Request permission

Abstract:

This paper is concerned with the Krull dimension (in the sense of Gabriel and Rentschler) of a differential operator ring $S[\theta ;\delta ]$, where $S$ is a right noetherian ring with finite Krull dimension $n$ and $\delta$ is a derivation on $S$. The main theorem states that $S[\theta ;\delta ]$ has Krull dimension $n$ unless there exists a simple right $S$-module $A$ such that $A{ \otimes _S}S[\theta ;\delta ]$ is not simple (as an $S[\theta ;\delta ]$-module) and $A$ has height $n$ in the sense that there exist critical right $S$-modules $A = {A_0},{A_1},\ldots ,{A_n}$ such that each ${A_i} \otimes _s S[\theta ;\delta ]$ is a critical $S[\theta ;\delta ]$-module, each ${A_i}$ is a minor subfactor of ${A_{i + 1}}$ and ${A_n}$ is a subfactor of $S$. If such an $A$ does exist, then $S[\theta ;\delta ]$ has Krull dimension $n + 1$. This criterion is simplified when $S$ is fully bounded, in which case it is shown that $S[\theta ;\delta ]$ has Krull dimension $n$ unless $S$ has a maximal ideal $M$ of height $n$ such that either ${\text {char(}}S/M) > 0$ or $\delta (M) \subseteq M$, and in these cases $S[\theta ;\delta ]$ has Krull dimension $n + 1$.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 275 (1983), 833-859
  • MSC: Primary 16A55; Secondary 16A05
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0682736-5
  • MathSciNet review: 682736