Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T04:10:27.399Z Has data issue: false hasContentIssue false
  • English
  • Français

Rings With Finite Maximal Invariant Subrings

Published online by Cambridge University Press:  20 November 2018

Charles Lanski
Charles Lanski*
Affiliation:
Department of Mathematics University of Southern California Los Angeles, CA 90089-1113 U.S.A. email: e-mail: clanski@mtha.usc.edu
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.

We prove that if φ is an (anti-) automorphism of a ring R with finite orbits on R, or integral over the integers, and if R contains a finite maximal φ-invariant subring, then R must be finite. Special cases are when φ has finite order or is an involution. Two corollaries are that R must be finite when R contains only finitely many φ-invariant subrings or has both ascending and descending chain conditions on φ invariant subrings. These generalize results in the literature for the special case when φ = idR.

Keywords

16P1016W2016P70
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 3 , 01 June 1996 , pp. 596 - 606
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Albert, A.A., Structure of Algebras, Amer. Math. Soc. Colloq. Publ. 24 (1961).Google Scholar
2. Bell, H.E. and Guerriero, F., Some conditions for finiteness and commutativity of rings, Internat. J. Math. Math. Sci. 13 (1990), 535–544.Google Scholar
3. Bell, H.E. and Klein, A.A., On finiteness of rings with finite maximal subrings, Internat. J. Math. Math. Sci. 16 (1993), 351–354.Google Scholar
4. Bergman, G.M. and Isaacs, I.M., Rings with fixed-point-free group actions, Proc. London Math. Soc. (3) 27 (1973), 69–87.Google Scholar
5. Gilmer, R., A note on rings with only finitely many subrings, Scripta Math. 29 (1973), 37–38.Google Scholar
6. Herstein, N., Noncommutative Rings, Cams Math. Monographs 15, Math. Assoc, of Amer. (1968).Google Scholar
7. Klein, A.A., The finiteness of a ring with a finite maximal subring, Comm. Algebra 21 (1993), 1389–1392.Google Scholar
8. Laffey, T.J., A finiteness theorem for rings, Proc. Roy. Irish Acad. 92A(1992), 285–288.Google Scholar
9. Lanski, C., Differential identities in prime rings with involution, Trans. Amer. Math. Soc. 291 (1985), 765–787.Google Scholar
10. Lanski, C., On the cardinality of rings with special subsets which are finite, Houston J. Math. 19 (1993), 357–373.Google Scholar
11. Martindale, W.S., III, Prime rings satisfying a generalized polynomial identity, J.Algebra 12 (1969), 576–584.Google Scholar
12. Montgomery, S., Fixed rings of finite automorphism groups of associative rings, Lecture Notes in Math, Springer-Verlag, New York, 818 (1980).Google Scholar
13. Szele, T., On a finiteness condition for modules, Publ. Math. Debrecen 3 (1954), 253–256.Google Scholar