Characterizations of semisimple classes by means of ring elements

Abstract

Radical classes defined by means of elements have been introduced in [17] and [14] and dual special radical classes have been also characterized by means of elements in [11]. We continue these ideas in a dual way, that is we characterize semisimple classes in terms of certain properties. In section 2 we prove that the class of P-rings is a semisimple class provided that property P satisfies conditions (b), (c) and (d). Moreover, for essentially closed semisimple classes such a property P can always be defined. The semisimple classes of the known concrete radicals is shown to be characterized easily in this way through several examples. In section 3 we characterize semisimple classes of supernilpotent, subidempotent and special radicals with some known examples. The last section is devoted to a characterization of semisimple classes of dual special radicals tailed by three corollaries showing that the semisimple classes of the Brown--McCoy, the Behrens and the antisimple radicals can be characterized in this way. All rings considered are associative. We shall write I <~ A when I is a nonzero ideal of a ring A and (S)A for the ideal of A generated by the subset S _c A. An ideal I of a ring A is called an essential ideal of A, denoted by I<~.A, if IAK~0 for every K~A. A ring is said to be semiprime if the intersection of all prime ideals is zero. A ring A is called subdirectly irreducible if its heart H -~ n (I i I <~ A) is nonzero. However, this heart is either an idempotent or a zero ring. A radical (class) will always mean a Kurosh--Amitsur radical (class). We shall use the upper radical and semisimple operators ~ and ~ acting on a class C of rings defined by