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On a class of radicals of rings

Part of: Rings and algebras arising under various constructions Radicals and radical properties of rings

Published online by Cambridge University Press:  09 April 2009

R. Mazurek
R. Mazurek
Affiliation:
Institute of Mathematics, University of Warsaw, Bialystok Division, Akademicka 2, 15-267 Bialystok, Poland, e-mail: mazurek@cksr.ac.bialystok.pl
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Abstract

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Let λ be a property that a lattice of submodules of a module may possess and which is preserved under taking sublattices and isomorphic images of such lattices and is satisfied by the lattice of subgroups of the group of integer numbers. For a ring R the lower radical Λ generated by the class λ(R) of R-modules whose lattice of submodules possesses the property λ is considered. This radical determines the unique ideal Λ (R) of R, called the λ-radical of R. We show that Λ is a Hoehnke radical of rings. Although generally Λ is not a Kurosh-Amitsur radical, it has the ADS-property and the class of Λ-radical rings is closed under extensions. We prove that Λ (Mn (R)) ⊆ Mn (Λ(R)) and give some illustrative examples.

MSC classification

Secondary: 16N99: None of the above, but in this section 16S90: Torsion theories; radicals on module categories 16S50: Endomorphism rings; matrix rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 59 , Issue 2 , October 1995 , pp. 184 - 192
Copyright
Copyright © Australian Mathematical Society 1995

References

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