Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-29T06:59:07.177Z Has data issue: false hasContentIssue false
  • English
  • Français

Extending Jordan Ideals and Jordan Homomorphisms of Symmetric Elements in a Ring with Involution

Published online by Cambridge University Press:  20 November 2018

Kirby C. Smith
Kirby C. Smith*
Affiliation:
University of Oklahoma, Norman, Oklahoma
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.

In this work, we show how the ideas in [3, pp. 6-12] can be used to give conditions under which Jordan ideals in the set of symmetric elements in an associative ring R with involution extend to associative ideals of R in a natural way. We also give conditions under which a Jordan homomorphism of the set of symmetric elements will extend to an associative homomorphism of R. Such work has been done on matrix rings with involution in [5; 6]. An abstract definition of a Jordan ring may be found in [3] as well as other background information.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 1 , February 1972 , pp. 50 - 59
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Cohn, P. M., Universal algebra (Harper and Row, New York, 1965).Google Scholar
2. Herstein, I. N., Lie and Jordan systems in simple rings with involution, Amer. J. Math. 78 (1956), 629–649.Google Scholar
3. Jacobson, N., Structure and representation of Jordan algebras, Amer. Math. Soc. Colloq. Publ., Vol. 39 (Amer. Math. Soc, Providence, R.I., 1968).Google Scholar
4. Jacobson, N. and Rickart, C. E., Jordan homomorphisms of rings, Trans. Amer. Math. Soc. 69 (1950), 479–502.Google Scholar
5. Jacobson, N. and Rickart, C. E., Homomorphisms of Jordan rings of self-adjoint elements, Trans. Amer. Math. Soc. 72 (1952), 310–322.Google Scholar
6. Martindale, W. S., III, Jordan homomorphisms of the symmetric elements of a ring with involution, J. of Algebra, 5 (1967), 232–249.Google Scholar
7. Osborn, J. M., Jordan and associative rings with nilpotent and invertible elements, J. of Algebra 15 (1970), 301–308.Google Scholar