On Modules and Radicals

Abstract

In the early sixties Andrunakievič and Rjabuhin extended the general theory of radicals for rings and groups to modules over associative rings ([1],[2]). As in the case of rings and groups in the work of Kuraš and Amitsur, the modules have to satisfy some axiomatic conditions in order to define an appropriate concept of radical, a so-called general class of modules. In this note we use their concepts of general class of modules with the corresponding radical class and develop it further. For any radical ℝ there exists a general class of modules ∑, such that the radical class corresponding to ∑, coincides with ℝ (Theorem 1). A radical ℝ (in the class of associative rings) is hereditary if for every ring A and any ideal B of A the equality ℝ(B) = B ∩ ℝ(A) holds. Analogous results hold for the ∑-radical of a class ∑ of modules (Propositions 8,9 and Theorem 2). For unexplained notions we refer to [1].