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ON RINGS WHOSE RIGHT ANNIHILATORS ARE BOUNDED

Published online by Cambridge University Press:  01 September 2009

SEO UN HWANG ,
NAM KYUN KIM  and
YANG LEE
SEO UN HWANG
Affiliation:
Department of Mathematics, Busan National University, Busan 609-735, Korea e-mail: [email protected]
NAM KYUN KIM
Affiliation:
College of Liberal Arts, Hanbat National University, Daejeon 305-719, Korea e-mail: [email protected]
YANG LEE
Affiliation:
Department of Mathematics Education, Busan National University, Pusan 609-735, Korea e-mail: [email protected]
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Abstract

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Jacobson said a a right ideal would be called bounded if it contained a non-zero ideal, and Faith said a ring would be called strongly right bounded if every non-zero right ideal were bounded. In this paper we introduce a condition that is a generalisation of strongly bounded rings and insertion-of-factors-property (IFP) rings, calling a ring strongly right AB if every non-zero right annihilator is bounded. We first observe the structure of strongly right AB rings by analysing minimal non-commutative strongly right AB rings up to isomorphism. We study properties of strongly right AB rings, finding conditions for strongly right AB rings to be reduced or strongly right bounded. Relating to Ramamurthi's question (i.e. Are right and left SF rings von Neumann regular?), we show that a ring is strongly regular if and only if it is strongly right AB and right SF, from which we may generalise several known results. We also construct more examples of strongly right AB rings and counterexamples to several naturally raised situations in the process.

Keywords

16D2516D4016E50
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 3 , September 2009 , pp. 539 - 559
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

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