Some Subidempotent Radicals

Abstract

This chapter discusses some of the sub-idempotent radicals. The basic definitions and results in radical theory may be found in Wiegandt and those for regular rings may be found in Goodearl. An example of a left hereditary radical that is not right hereditary was given. It was asked whether such an example could exist among radical classes consisting only of idempotent rings. Beider has shown that such a radical class is left hereditary if it consists of rings that are von Neumann regular and contain no non-zero nilpotent element. As these properties are left-right symmetric, it follows that all such radicals are right hereditary. A ring R that is von Neumann regular and contains no non-zero nilpotent element is referred to in the literature as strongly regular or abelian regular. The following conditions on a ring R are equivalent: (1) R is strongly regular, (2) every left accessible subring of R is idempotent, (3) every right accessible subring of R is idempotent, and (4) R is semi-prime and every prime image of R is a division ring.