Radicals of rings and pullbacks

AbstractGeneralizing 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

A simplicial approach to factorization systems and Kurosh–Amitsur radicals

AbstractRegarding categories as simplicial sets via the nerve functor, we extend the notion of a factorization system from morphisms in a category, to 1-simplexes in an arbitrary simplicial set. Applied to what we call the simplicial set of short exact sequences, it gives the notion of Kurosh–Amitsur radical. That is, we present a unified approach to factorization systems and radicals

NON-POINTED EXACTNESS, RADICALS, CLOSURE OPERATORS

In this paper it is shown how nonpointed exactness provides a framework which allows a simple categorical treatment of the basics of Kurosh-Amitsur radical theory in the nonpointed case. This is made possible by a new approach to semi-exactness, in the sense of the first author, using adjoint functors. This framework also reveals how categorical closure operators arise as radical theories