RADICALS OF RINGS AND PULLBACKS

Generalizing various concrete radicals in associative rings like the n ilradical, the Jacobson radical, and so on, A.G. Kurosh and S.A. Amits ur introduced an abstract notion of radical in the early 1950s. The ba sic notions of their general radical theory can be characterized by pr operties which are ''almost'' categorical - in the sense that they can be conveniently defined in the category of rings or even in suitable categories of OMEGA-groups but not in general categories. Here we are going to characterize radicals of associative rings by means of pullba cks, a notion which is of a purely categorical nature. Throughout the paper we shall work in the category C of associative rings (not necess arily with identity), just calling them ''rings''. We hope that our tw o categorical characterizations of semisimple classes in C can provide natural general frameworks for radical theory, just as localizations do for torsion theories.