Characterizations of semisimple classes

Abstract

Abstract Two characterizations of semisimple classes of associative and alternative rings (and semigroups with 0) are given: (i) A class is a semisimple class if and only if it is hereditary, closed under extensions and subdirect sums; (ii) A class is a semisimple class if and only if it is hereditary, closed under extensions, and has the co-inductive property. The first characterization sharpens Armendariz's (1968) result proved for associative rings, the second one is categorically dual to a characterization of radical classes due to Amitsur (1954).