zbMATH — the first resource for mathematics

Iterated skew polynomial rings of Krull dimension two. (English) Zbl 0809.16031
Let $$A$$ be a finitely generated algebra over an algebraically closed field $$k$$. Let $$\alpha$$ be a $$k$$-automorphism and let $$R$$ denote the iterated skew polynomial ring $$A[y; \alpha][x; \alpha^{-1}, \delta]$$. The main intent of this article is to show that if $$A$$ is $$\alpha$$-simple and a principal ideal domain and if $$X$$ is a finitely generated right $$R$$-module such that both $$X_ x$$ and $$X_ y$$ have finite length, then $$X$$ has finite length. This eliminates the need in earlier work of the author to assume that all finite dimensional modules were semisimple.

MSC:
 16S36 Ordinary and skew polynomial rings and semigroup rings 16P60 Chain conditions on annihilators and summands: Goldie-type conditions 16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras 16P10 Finite rings and finite-dimensional associative algebras
Full Text:
References:
 [1] D. A. Jordan, Iterated skew polynomial rings and quantum groups. J. Algebra156, 194-218 (1993). · Zbl 0809.16032 · doi:10.1006/jabr.1993.1070 [2] D. A. Jordan, Krull and global dimension of certain iterated skew polynomial rings. In: Abelian Groups and Noncommutative Rings, a Collection of Papers in Memory of Robert B. Warfield, Jr., Contemporary Mathematics. Amer. Math. Soc.130, 201-213 (1993). [3] J. C.McConnell and J. C.Robson, Noncommutative Noetherian rings. Chichester 1987. · Zbl 0644.16008 [4] S. P. Smith, Krull dimension of the enveloping algebra ofsl(2, ?). J. Algebra71, 189-194 (1981). · Zbl 0468.17003 · doi:10.1016/0021-8693(81)90114-9 [5] S. P. Smith, A class of algebras similar to the enveloping algebra ofsl(2). Trans. Amer. Math. Soc.322, 285-314 (1990). · Zbl 0732.16019 · doi:10.2307/2001532
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.