Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-12-02T20:41:51.628Z Has data issue: false hasContentIssue false
  • English
  • Français

The Coefficient Ring of a Primitive Group Ring

Published online by Cambridge University Press:  20 November 2018

John Lawrence
John Lawrence*
Affiliation:
McGill University, Montreal, Quebec; Carleton University, Ottawa, Ontario
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.

All rings are associative with unity. A ring R is prime if xRy ≠ 0 whenever x and y are nonzero. A ring R is (left) primitive if there exists a faithful irreducible left R-module.

If the group ring R[G] is primitive, what can we say about R? First, since every primitive ring is prime, we know that R is prime, by the following

THEOREM 1 (Connell [1, 675]). The group ring R[G] is prime if and only if R is prime and G has no non-trivial finite normal subgroup.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 3 , 01 June 1975 , pp. 489 - 494
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Connell, I., On the group ring, Can. J. Math. 15 (1963), 650–685.Google Scholar
2. Fisher, J. and Snider, R., Prime von Neumann regular rings and primitive group algebras, Proc. Amer. Math. Soc. U (1974), 244–250.Google Scholar
3. Formanek, E., Group rings of free products are primitive, J. Algebra 26 (1973), 508–511.Google Scholar
4. Goodearl, K., Prime ideals in regular self-infective rings, Can. J. Math. 25 (1973), 829–839.Google Scholar
5. Goodearl, K. and Handelman, D., Simple self-infective rings (to appear).Google Scholar
6. Goodearl, K., Handelman, D., and Lawrence, J., Strongly prime and completely torsion-free rings, Carleton Math. Series.Google Scholar
7. Handelman, D. and Lawrence, J., Strongly prime rings, Trans. Amer. Math. Soc. (to appear).Google Scholar
8. Kaplansky, I., Algebraic and analytic aspects of operator algebras (Amer. Math. Soc).Google Scholar
9. Lambek, J., Lectures on rings and modules (Ginn and Blaisdell, New York, 1966).Google Scholar
10. Osofsky, B., A non-trivial ring with non-rational infective hull, Can. Math. Bull. 10 (1967), 275–282.Google Scholar