Radicals of rings and pullbacks

Abstract

Generalizing various concrete radicals in associative rings like the nilradical, the Jacobson radical, and so on, A.G. Kurosh and S.A. Amitsur introduced an abstract notion of radical in the early 1950s. The basic notions of their general radical theory can be characterized by properties which are “almost” categorical – in the sense that they can be conveniently defined in the category of rings or even in suitable categories of Ω-groups but not in general categories. Here we are going to characterize radicals of associative rings by means of pullbacks, a notion which is of a purely categorical nature. Throughout the paper we shall work in the category C of associative rings (not necessarily with identity), just calling them “rings”. We hope that our two categorical characterizations of semisimple classes in C can provide natural general frameworks for radical theory, just as localizations do for torsion theories.