SEMICOPRIME PRERADICALS

We define the concept of "semicoprime" for preradicals and for submodules and we prove some properties that relate both of them. For any ring we define the ultrasocle preradical as a certain join of maximal semicoprime preradicals. It defines a kind of primary decomposition on modules. We compare the greatest semicoprime preradical, the meet of all unipotent preradicals, the socle preradical, and the ultrasocle preradical. We characterize rings which are finite product of simple rings in terms of some of these preradicals. We study the least semicoprime preradical above any preradical and we prove some of its properties. Using "Amitsur constructions" we define some related operators and prove some of their properties.