Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-15T16:35:17.487Z Has data issue: false hasContentIssue false
  • English
  • Français

On smallest radical and semi-simple classes

Published online by Cambridge University Press:  18 May 2009

W. G. Leavitt  and
J. F. Watters
W. G. Leavitt
Affiliation:
University of Nebraska, Lincoln, Nebraska, U.S.A.
J. F. Watters
Affiliation:
The University, Leicester
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 a recent paper [5] one of us has given a sufficient condition to be satisfied by a given property of radical classes within a universal class w in order that, for any subclass ℳ of w, there should be a smallest radical class having the given property and containing ℳ. The sufficient condition is that the classof all radical classes with the given property can be characterised as the class of all radical classes fixed by an admissible function F (see Section 1 below). In this paper a necessary and sufficient condition is derived and the corresponding result for semi-simpleclasses is also presented. These results are given in Section 2.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 12 , Issue 2 , September 1971 , pp. 98 - 104
Copyright
Copyright © Glasgow Mathematical Journal Trust 1971

References

REFERENCES

1.Divinsky, N., Rings and Radicals (London, 1965).Google Scholar
2.Leavitt, W. G., Sets of radical classes, Publ. Math. Debrecen 14 (1967), 321–324.CrossRefGoogle Scholar
3.Leavitt, W. G., Strongly hereditary radicals, Proc. Amer. Math. Soc. 21 (1969), 703–705.CrossRefGoogle Scholar
4.Leavitt, W. G., Hereditary semi-simple classes, Glasgow Math. J. 11 (1970), 7–8.CrossRefGoogle Scholar
5.Leavitt, W. G., Radical and semi-simple classes with specified properties, Proc. Amer. Math. Soc. 24 (1970), 680–687.Google Scholar
6.Rjabuhin, Ju. M., On the imbedding of radicals, Isv. Akad. Nauk. Mold. SS 11 (1963), 34–40.Google Scholar
7.Rjabuhin, Ju. M., Lower radicals of rings, Mat. Zametki 2 (1967), 239–244.Google Scholar