Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-14T19:20:12.050Z Has data issue: false hasContentIssue false
  • English
  • Français

Essentially closed radical classes

Published online by Cambridge University Press:  09 April 2009

N. V. Loi
N. V. Loi
Affiliation:
L Eötvös UniversityBudapestHungary
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.

The main goal of this paper is to describe radical classes closed under essential extensions. It turns out that such classes are precisely the homomorphically closed semisimple classes, and hence a radical class is essentially closed if and only if it is subdirectly closed. Moreover, a class is closed under homomorphic images, direct sums and essential extensions if and only if it is an essentially closed radical class. Also radical classes are investigated which are closed under Dorroh essentially extensions only, such a radical class R consists of idempotent rings provided that R does not contain the ring of integers, meanwhile all the other radicals satisfy this requirement. A description of (hereditary and) Dorroh essentially closed radicals is given in Theorem 4.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 35 , Issue 1 , August 1983 , pp. 132 - 142
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Armendariz, E. P., ‘Closure properties in radical theory’, Pacific J. Math. 26 (1968) 1–7.CrossRefGoogle Scholar
[2]Gardner, B. J. and Stewart, P. N., ‘On semisimple radical classes’, Bull. Austral. Math. Soc. 13 (1975) 349–353.CrossRefGoogle Scholar
[3]van Leeuwen, L. C. A., ‘Properties of semisimple classes’, J. Natur. Sci. and Math. 15 (1975) 59–67.Google Scholar
[4]Stewart, P. N., ‘Semisimple radical classes’, J. Math. 32 (1970) 249–254.Google Scholar
[5]Szász, F. A., ‘A second almost subidempotent radical for rings’, Math. Nachr. 66 (1975) 283–289.CrossRefGoogle Scholar
[6]Wiegandt, R., Radical and semisimple classes of rings (Queen's Papers in Pure and Appl. Math. 37 Kingston 1974).Google Scholar