Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T11:53:40.130Z Has data issue: false hasContentIssue false
  • English
  • Français

Prime rings for which the set of nonzero ideals is a special class

Published online by Cambridge University Press:  09 April 2009

B. J. Gardner
B. J. Gardner
Affiliation:
Department of Mathematics University of TasmaniaG.P.O. Box 252C Hobart Tasmania 7001, Australia
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 show that the rings described in the title are precisely the indecomposable injectives for the category whose objects are the associative rings and whose morphisms are the ring homomorphisms with accessible images. These rings are more or less completely known. Those of cardinality greater than that of the continuum are subdirectly irreducible but there are some nontrivial principal ideal domains in the class.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 51 , Issue 1 , August 1991 , pp. 27 - 32
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Andrunakievič, V. A., ‘Radicals of associative rings I’, Amer. Math. Soc. Transl. (2) 52 (1966), 95–128.Google Scholar
[2]Andrunakievich, V. A. and Ryabukhin, Yu. M., Radicals of algebras and structure theory (in Russian), Nauka, Moscow, 1979.Google Scholar
[3]Gardner, B. J., ‘Injectives for ring monomorphisms with accessible images’, Comm. Algebra 10 (1982), 673–694.CrossRefGoogle Scholar
[4]Gardner, B. J. and Stewart, P. N., ‘Injectives for ring monomorphisms with accessible images, II’, Comm. Algebra 13 (1985), 133–145.CrossRefGoogle Scholar
[5]Heyman, G. A. P. and Roos, C., ‘Essential extensions in radical theory’, J. Austral. Math. Soc. (Series A) 23 (1977), 340–347.CrossRefGoogle Scholar
[6]Ryabukhin, Yu. M., ‘Supernilpotent and special radicals’, (in Russian), Mat. Issled. 48 (1978), 80–93.Google Scholar