Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-19T01:45:30.688Z Has data issue: false hasContentIssue false

The semicentre of a group algebra

Published online by Cambridge University Press:  20 January 2009

Paul Wauters
Affiliation:
Department of Mathematics, Limburgs Universitair Centrum, Diepenbeek, Belgium, E-mail address: pwauters@luc.ac.be
Rights & Permissions [Opens in a new window]

Extract

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.

We study the semicentre of a group algebra K[G] where K is a field of characteristic zero and G is a polycyclic-by-finite group suchthat Δ(G) is torsion-free abelian. Several properties about the structure of this ring are proved, in particular as to when is the semicentre a UFD. Examples are constructed when this is not the case. We also prove necessary and sufficient conditions for every normal element of K[G] which belongs to K[Δ(G)] to be the product of a unit and a semi-invariant.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1999

References

REFERENCES

1.Abbasi, G. Q., Kobayashi, S., Marubayashi, H., and Ueda, A., Noncommutative unique factorization rings, Comm. Algebra 19 (1991), 167198.Google Scholar
2.Anderson, D. F., Graded Krull domains, Comm. Algebra 7 (1979), 79106.CrossRefGoogle Scholar
3.Brown, K. A., Height one primes of polycyclic group rings, J. London Math. Soc. 32 (1985), 426438.CrossRefGoogle Scholar
4.Brown, K. A., Class groups and automorphism groups of group rings, Glasgow Math. J. 28 (1986), 7986.CrossRefGoogle Scholar
5.Brown, K. A. and Lorenz, M., Grothendieck groups of invariant rings and of group rings, J. Algebra 166 (1994), 423454.Google Scholar
6.Chatters, A. W. and Jordan, D. A., Non-commutative unique factorization rings, J. London Math. Soc 33 (1986), 2232.CrossRefGoogle Scholar
7.Delvaux, L., Nauwelaerts, E. and Ooms, A. I., On the semi-centre of a universal enveloping algebra, J. Algebra 94 (1985), 324346.CrossRefGoogle Scholar
8.Farkas, D. F., Multiplicative invariants, Enseign. Math. 30 (1984), 141157.Google Scholar
9.Le Bruyn, L. and Ooms, A. I., The semicenter of an enveloping algebra is factorial, Proc. Amer. Math. Soc. 93 (1985), 397400.CrossRefGoogle Scholar
10.Lorenz, M., Class groups of multiplicative invariants, J. Algebra 177 (1995), 242254.CrossRefGoogle Scholar
11.Lorenz, M. and Passman, D. S., Centers and prime ideals in group algebras of polycyclic-by-finite groups, J. Algebra 57 (1979), 355386.CrossRefGoogle Scholar
12.Malliavin, M. P., Ultraproduit d'algèbres de Lie (LNM 924, 1982), 157166.Google Scholar
13.Moeglin, C., Factorialité dans les algèbres enveloppantes, C. R. Acad. Sci. Paris Ser. A 282 (1976), 12691272.Google Scholar
14.Moeglin, C., Idéaux bilatères dans les algèbres enveloppantes, Bull. Soc. Math. France 108 (1980), 143186.CrossRefGoogle Scholar
15.Montgomery, S. and Passman, D. S., X-inner automorphisms of group rings, Houston J. Math. 7 (1981), 395402.Google Scholar
16.Montgomery, S. and Passman, D. S., X-inner automorphisms of group rings II, Houston J. Math. 8 (1982), 537544.Google Scholar
17.Nauwelaerts, E. and Ooms, A. I., Weights of semi-invariants of the quotient division ring of an enveloping algebra, Proc. Amer. Math. Soc. 104 (1988), 1319.Google Scholar
18.Ooms, A. I. and Wauters, P., Primitive extensions of an enveloping algebra, J Algebra 161 (1993), 392405.Google Scholar
19.Passman, D. S., The algebraic structure of group rings (Wiley-Interscience, New York, 1977).Google Scholar
20.Passman, D. S., Computing the symmetric ring of quotients, J. Algebra 105 (1987), 207235.Google Scholar
21.Passman, D. S., Infinite crossed products (Academic Press Inc., San Diego, 1989).Google Scholar
22.Smith, M., Semi-invariant rings, Comm. Algebra 13 (1985), 12831298.Google Scholar