Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-24T08:25:37.428Z Has data issue: false hasContentIssue false
  • English
  • Français

Ideals in simple rings

Published online by Cambridge University Press:  09 April 2009

W. Harold Davenport
W. Harold Davenport
Affiliation:
Department of Mathematics, University of Petroleum & Minerals, Dhahran, Saudi Arabia.
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.

In this article, we define the concept of a Malcev ideal in an alternative ring in a manner analogous to Lie ideals in associative rings. By using a result of Kleinfield's we show that a nonassociative alternative ring of characteristic not 2 or 3 is a ring sum of Malcev ideals Z and [R, R] where Z is the center of R and [R, R] is a simple non-Lie Malcev ideal of R. If R is a Cayley algebra over a field F of characteristic 3 then [R, R] is a simple 7 dimensional Lie algebra. A similar result is obtained if R is a simple associative ring.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 25 , Issue 3 , May 1978 , pp. 322 - 327
Copyright
Copyright © Australian Mathematical Society 1978

References

Davenport, W. Harold (1971), The killing form on Malcev algebras (University of Alabama dissertation).Google Scholar
Herstein, I. N. (1969), Topics in Ring Theory (The University of Chicago Press).Google Scholar
Jacobson, N. and Rickart, C. E. (1950), ‘Jordan homomorphism of rings’, Trans. Amer. Math. Soc. 69, 479–502.CrossRefGoogle Scholar
Erwin, Kleinfield (1957), ‘Alternative nil rings’, Annals of Math. 66, 395–399.Google Scholar
Lam, T. Y. (1973), The Algebraic Theory of Quadratic Forms (W. A. Benjamin, Inc., Reading, Mass.).Google Scholar
Malcev, A. I. (1955), ‘Analytic loops’, Mat. Sb., 78, 569–578.Google Scholar
Sagle, A. A. (1961), ‘Malcev algebras’, Trans. Amer. Math. Soc. 101, 426–458.CrossRefGoogle Scholar
Schafer, R. D. (1966), An introduction to nonassociative algebras, Pure and Appl. Math., 22 (Academic Press, New York).Google Scholar