Strongly prime near-rings 2

Abstract

by N. J. GROENEWALD(Received 24th October 1985, revised 22nd July 1987)1. IntroductionStrongly prime rings were introduced by Handelman and Lawrence [5] and in [2]Groenewald and Heyman investigated the upper radical determined by the class of allstrongly prime rings. In this paper we extend the concept of strongly prime to near-rings. We show that the class M of distributively generated near-rings is a special classin the sense of Kaarli [6]. We also show that if N is any distributively generated near-ring, then UM(N), UM denotes the upper radical determined by the class M, coincideswith the intersection of all the strongly prime ideals of N.2. PreliminariesUnless otherwise stated, all near-rings are zero-symmetric right near-rings. Forundefined terminologies, we refer to [9].Definition 1. Let N be a near-ring. N {right) is calle stronglyd prime if and only if forevery O^aeN there exists a finite subset F of N r(aF) such = tha {neN:aFn=O}t = 0. Fis called an insulator of a in N.We now give the following alternative definition (c.f. [8] for corresponding definitionfor rings).Definition 2. Let N be a near-ring. N is called (right) strongly prime if and only ifevery nonzero ideal I of N contains a finite subset F 0. such that r(F) =The two definitions of strongly prime agree for the class of zero-symmetric near-rings.The proof of this is based on the following observation and lemma.Observation. Let N be a zero-symmetric near-ring and XcN. The ideal generatedby X is the intersection of all ideals containing X and can be obtained as follows: