ON RINGS WHOSE RIGHT ANNIHILATORS ARE BOUNDED

摘要

Jacobson said a a right ideal would be called bounded if it containedna non-zero ideal, and Faith said a ring would be called strongly right bounded if everynnon-zero right ideal were bounded. In this paper we introduce a condition that is angeneralisation of strongly bounded rings and insertion-of-factors-property (IFP) rings,ncalling a ring strongly right AB if every non-zero right annihilator is bounded.We firstnobserve the structure of strongly rightABrings by analysing minimal non-commutativenstrongly right AB rings up to isomorphism. We study properties of strongly right ABnrings, finding conditions for strongly right AB rings to be reduced or strongly rightnbounded. Relating to Ramamurthi’s question (i.e. Are right and left SF rings vonnNeumann regular?), we show that a ring is strongly regular if and only if it is stronglynright AB and right SF, from which we may generalise several known results. We alsonconstruct more examples of strongly right AB rings and counterexamples to severalnnaturally raised situations in the process.