×

zbMATH — the first resource for mathematics

Height one prime ideals of certain iterated skew polynomial rings. (English) Zbl 0804.16028
The author studies the height-1 prime ideals of the ring \(R\) described below. This very interesting but rather complicated paper contains much more than is covered by the following summary. Let \(A\) be a commutative integral domain which is finitely-generated as an algebra over an algebraically-closed field \(k\); let \(\alpha\) be a \(k\)-automorphism of \(A\); and let \(u\) and \(\rho\) be fixed non-zero elements of \(A\) and \(k\) respectively. The ring \(R\) is generated over \(A\) by \(x\) and \(y\) subject to the relations: \(xy - \rho yx = u - \rho \alpha(u)\); \(xa = \alpha^{- 1}(a) x\) and \(ya = \alpha(a) y\) for all \(a\) in \(A\). Alternatively, it can be constructed in two stages as an iterated skew polynomial ring in \(x\) and \(y\) over \(A\). The author previously studied \(R\) when \(\rho = 1\), but this new wider context includes additional important examples such as the quantized Weyl algebra. Many of the results include the further assumptions that \(A \neq k\) and that \(A\) is \(\alpha\)-simple, and we shall make these assumptions from now on. A complete description is given of all the height-1 prime ideals of \(R\), together with generators for those which are principal. All the principal height-1 primes of \(R\) are shown to be primitive (with corresponding simple modules constructed explicitly), and if \(A\) has Krull dimension 1 then every height-1 prime of \(R\) is principal and primitive.

MSC:
16S36 Ordinary and skew polynomial rings and semigroup rings
16D25 Ideals in associative algebras
16U20 Ore rings, multiplicative sets, Ore localization
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1112/jlms/s2-9.2.337 · Zbl 0294.16019 · doi:10.1112/jlms/s2-9.2.337
[2] DOI: 10.1016/S0021-8693(05)80036-5 · Zbl 0779.16010 · doi:10.1016/S0021-8693(05)80036-5
[3] DOI: 10.1112/jlms/s2-33.1.22 · Zbl 0601.16001 · doi:10.1112/jlms/s2-33.1.22
[4] Chatters, Math. Proc. Cambridge Philos. Soc 95 pp 49– (1984)
[5] Zariski, Commutative Algebra I (1958) · Zbl 0322.13001
[6] DOI: 10.2977/prims/1195176848 · Zbl 0676.46050 · doi:10.2977/prims/1195176848
[7] Seidenberg, Pacific J. Math 16 pp 167– (1966) · Zbl 0133.29202 · doi:10.2140/pjm.1966.16.167
[8] McConnell, Noncommutative Noetherian Rings (1987) · Zbl 0644.16008
[9] Kaplansky, Fields and Rings (1969)
[10] Jordan, Math. Z
[11] Jordan, Finite-dimensional simple modules over certain iterated skew polynomial rings · Zbl 0829.16017 · doi:10.1016/0022-4049(94)00026-F
[12] Jordan, Amer. Math. Soc 130 pp 201– (1992)
[13] DOI: 10.1006/jabr.1993.1070 · Zbl 0809.16032 · doi:10.1006/jabr.1993.1070
[14] Jordan, Glasgow Math. J 19 pp 79– (1978)
[15] Cohn, Free Rings and their Relations (1985)
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.