Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T19:33:01.644Z Has data issue: false hasContentIssue false
  • English
  • Français

INJECTIVE MODULES OVER DOWN-UP ALGEBRAS

Published online by Cambridge University Press:  24 June 2010

PAULA A. A. B. CARVALHO ,
CHRISTIAN LOMP  and
DILEK PUSAT-YILMAZ
PAULA A. A. B. CARVALHO
Affiliation:
Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, 4169-007 Porto, Portugal
CHRISTIAN LOMP
Affiliation:
Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, 4169-007 Porto, Portugal
DILEK PUSAT-YILMAZ
Affiliation:
Department of Mathematics, Izmir Institute of Technology, 35430 Gulbahce, Urla-Izmir, Turkey
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this paper is to study finiteness conditions on injective hulls of simple modules over Noetherian down-up algebras. We will show that the Noetherian down-up algebras A(α, β, γ) which are fully bounded are precisely those which are module-finite over a central subalgebra. We show that injective hulls of simple A(α, β, γ)-modules are locally Artinian provided the roots of X2 − αX − β are distinct roots of unity or both equal to 1.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue A: Rings and Modules in Honour of Patrick F. Smith's 65th Birthday , July 2010 , pp. 53 - 59
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Bavula, V. V. and Lenagan, T. H., Generalized Weyl algebras are tensor Krull minimal, J. Algebra 239 (1) (2001), 93–111.CrossRefGoogle Scholar
2.Benkart, G. and Roby, T., Down-up algebras, J. Algebra 209 (1) (1998), 305–344.CrossRefGoogle Scholar
3.Carvalho, P. A. A. B., On the prime spectra of some Noetherian rings, PhD Thesis (University of Glasgow, 1998).Google Scholar
4.Carvalho, P. A. A. B. and Musson, I. M., Down-up algebras and their representation theory, J. Algebra 228 (2000), 286–310.CrossRefGoogle Scholar
5.Cauchon, G., Anneaux de polynòmes essentiellement borns, in Ring theory (Proc. Antwerp Conf. (NATO Adv. Study Inst.), University of Antwerp, Antwerp, 1978), Lecture Notes in Pure and Applied Mathematics, vol. 51 (van Oystaeyen, F., Editor) (Dekker, 1979), 27–42.Google Scholar
6.Dahlberg, R. L., Injective hulls of simple sl(2, C) modules are locally Artinian, Proc. Amer. Math. Soc. 107 (1) (1989), 35–37.Google Scholar
7.Goodearl, K. R. and Warfield, R. B. Jr., An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts 16. (Cambridge University Press, Cambridge, 1989).Google Scholar
8.Jans, J. P., On co-Noetherian rings, J. Lond. Math. Soc. 1 (2) (1969), 588–590.CrossRefGoogle Scholar
9.Jategaonkar, A. V., Jacobson's conjecture and modules over fully bounded Noetherian rings, J. Algebra 30 (1974), 103–121.CrossRefGoogle Scholar
10.Jategaonkar, A. V., in Certain injectives are Artinian. in Noncommutative ring theory (Internat. Conf., Kent State University, Kent Ohio, 1975). Lecture Notes in Mathematics, vol. 545 (Springer, Berlin, 1976) 128–139.Google Scholar
11.Kirkman, E., Musson, I. M. and Passman, D. S., Noetherian down-up algebras, Proc. Amer. Math. Soc. 127 (11) (1999), 3161–3167.CrossRefGoogle Scholar
12.Krause, G., On fully left bounded left Noetherian Rings, J. Algebra 23 (1972), 88–99.CrossRefGoogle Scholar
13.Kulkarni, R. S., Down-up algebras and their Representations, J. Algebra 245 (2001), 431–462.CrossRefGoogle Scholar
14.Kulkarni, R. S., Down-up algebras at roots of unity, Proc. Amer. Math. Soc. 136 (10) (2008), 3375–3382.CrossRefGoogle Scholar
15.Matlis, E., Injective modules over Noetherian rings, Pac. J. Math. 8 (1958), 511–528.CrossRefGoogle Scholar
16.Matlis, E., Modules with descending chain condition, Trans. Amer. Math. Soc. 97 (3) (1960), 495–508.CrossRefGoogle Scholar
17.McConnell, J. C. and Robson, J. C., Noncommutative Noetherian rings, Grad. Stud. Math. 30, AMS (2000).Google Scholar
18.Musson, I. M., Some examples of modules over Noetherian rings, Glasgow Math. J. 23 (1982), 9–13.CrossRefGoogle Scholar
19.Praton, I., Primitive ideals in Noetherian down-up algebras, Commun. Algebra 32 (2) (2004), 443–471.CrossRefGoogle Scholar
20.Schelter, W., Essential extensions and intersection theorems, Proc. Amer. Math. Soc. 53 (2) (1975), 328–330.CrossRefGoogle Scholar
21.Vamos, P., The dual of the notion of “finitely generated”, J. Lond. Math. Soc. 43 (1968), 643–646.CrossRefGoogle Scholar
22.Zhao, K., Centers of down-up algebras, J. Algebra 214 (1999), 103–121.CrossRefGoogle Scholar