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Transactions of the American Mathematical Society

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Krull dimension of differential operator rings. III. Noncommutative coefficients


Authors: K. R. Goodearl and T. H. Lenagan
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
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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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DOI: https://doi.org/10.1090/S0002-9947-1983-0682736-5
Keywords: Krull dimension, differential operator ring, Ore extension, noetherian ring
Article copyright: © Copyright 1983 American Mathematical Society