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Transactions of the American Mathematical Society

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



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
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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Keywords: Krull dimension, differential operator ring, Ore extension, noetherian ring
Article copyright: © Copyright 1983 American Mathematical Society