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 $n + 1$.

- Jan-Erik Björk,
*The global homological dimension of some algebras of differential operators*, Invent. Math.**17**(1972), 67–78. MR**320078**, DOI https://doi.org/10.1007/BF01390024 - Walter Borho, Peter Gabriel, and Rudolf Rentschler,
*Primideale in Einhüllenden auflösbarer Lie-Algebren (Beschreibung durch Bahnenräume)*, Lecture Notes in Mathematics, Vol. 357, Springer-Verlag, Berlin-New York, 1973 (German). MR**0376790** - John Cozzens and Carl Faith,
*Simple Noetherian rings*, Cambridge University Press, Cambridge-New York-Melbourne, 1975. Cambridge Tracts in Mathematics, No. 69. MR**0396660** - K. R. Goodearl,
*Global dimension of differential operator rings. II*, Trans. Amer. Math. Soc.**209**(1975), 65–85. MR**382359**, DOI https://doi.org/10.1090/S0002-9947-1975-0382359-7 - K. R. Goodearl and T. H. Lenagan,
*Krull dimension of differential operator rings. IV. Multiple derivations*, Proc. London Math. Soc. (3)**47**(1983), no. 2, 306–336. MR**703983**, DOI https://doi.org/10.1112/plms/s3-47.2.306 - K. R. Goodearl and R. B. Warfield Jr.,
*Krull dimension of differential operator rings*, Proc. London Math. Soc. (3)**45**(1982), no. 1, 49–70. MR**662662**, DOI https://doi.org/10.1112/plms/s3-45.1.49 - Kenneth R. Goodearl and Robert B. Warfield Jr.,
*Primitivity in differential operator rings*, Math. Z.**180**(1982), no. 4, 503–523. MR**667005**, DOI https://doi.org/10.1007/BF01214722 - Robert Gordon and J. C. Robson,
*Krull dimension*, American Mathematical Society, Providence, R.I., 1973. Memoirs of the American Mathematical Society, No. 133. MR**0352177** - Robert Hart,
*Krull dimension and global dimension of simple Ore-extensions*, Math. Z.**121**(1971), 341–345. MR**297759**, DOI https://doi.org/10.1007/BF01109980 - T. Hodges and J. C. McConnell,
*On Ore and skew-Laurent extensions of Noetherian rings*, J. Algebra**73**(1981), no. 1, 56–64. MR**641634**, DOI https://doi.org/10.1016/0021-8693%2881%2990348-3 - Nathan Jacobson,
*The Theory of Rings*, American Mathematical Society Mathematical Surveys, Vol. II, American Mathematical Society, New York, 1943. MR**0008601** - Arun Vinayak Jategaonkar,
*Jacobson’s conjecture and modules over fully bounded Noetherian rings*, J. Algebra**30**(1974), 103–121. MR**352170**, DOI https://doi.org/10.1016/0021-8693%2874%2990195-1 - T. H. Lenagan,
*Krull dimension of differential operator rings. II. The infinite case*, Rocky Mountain J. Math.**13**(1983), no. 3, 475–480. MR**715770**, DOI https://doi.org/10.1216/RMJ-1983-13-3-475 - Y. Nouazé and P. Gabriel,
*Idéaux premiers de l’algèbre enveloppante d’une algèbre de Lie nilpotente*, J. Algebra**6**(1967), 77–99 (French). MR**206064**, DOI https://doi.org/10.1016/0021-8693%2867%2990015-4 - G. Renault,
*Algèbre non commutative*, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1975 (French). Collection “Varia Mathematica”. MR**0384845** - J. C. Robson,
*Idealizers and hereditary Noetherian prime rings*, J. Algebra**22**(1972), 45–81. MR**299639**, DOI https://doi.org/10.1016/0021-8693%2872%2990104-4 - Alex Rosenberg and J. T. Stafford,
*Global dimension of Ore extensions*, Algebra, topology, and category theory (a collection of papers in honor of Samuel Eilenberg), Academic Press, New York, 1976, pp. 181–188. MR**0409564**
C. L. Wangneo,

*Polynomial rings over certain classes of rings*(to appear).

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Keywords:
Krull dimension,
differential operator ring,
Ore extension,
noetherian ring

Article copyright:
© Copyright 1983
American Mathematical Society