Three Distinct Non-Hereditary Radicals Which Coincide with the Classical Radical for Rings with D.C.C.

Abstract

Majumdar and Paul [3] introduced and studied a new radical E defined as the upper radical determined by the class of all rings each of whose ideals is idempotent. In this paper the authors continue the study further and also study the join radical and the intersection radical (due to Leavitt) obtained from E and the Jacobson radical J. These have been denoted by E + J and EJ respectively. The radical and the semisimple rings corresponding to E + J and EJ have been obtained. Both of these radicals coincide with the classical nil radical for Artinian rings. Important properties of these radicals and their position among the well-known special radicals have been investigated. It has been proved that E, EJ and E + J are non-hereditary. It has also been proved as an independent result that the nil radical N is not dual, i.e., N ≠ N  .