Hereditariness of strong and stable radicals

Abstract

The aim of this paper is to discuss some relations among hereditary, strong and stable radicals. In particular we investigate hereditariness of lower strong and stable radicals. Some facts obtained are related to some results and questions of [2,6,7].All rings in the paper are associative. Fundamental definitions and properties of radicals may be found in [9]. Definitions of hereditary and strong radicals are used as in Sands [7]. We say that a radical S is left {right) stable if (p):for every ring R and every left (right) ideal / of R it follows S(I)QS(R).Of course any left (right) stable radical is left (right) strong. If P is a subring of a ring R then P* will denote the ideal of R generated by P. Of course P* = JR 1 PR 1 where R 1 is the natural extension of R to the ring with unity. LEMMA 1. Let S be a radical and P a subring of a ring R such that P°eS. Then (P*)°eS. In particular if S is left strong (stable) then so is S°.Proof. Since P* = R

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