ON STRONGLY RIGHT BOUNDED AND RIGHT COMPLEMENT BOUNDED NEAR-RINGS.

Abstract

This work deals mainly with translating the known notion in ring theory of bounding one sided ideals by two sided ones (Jacobson [18]) to that of near-ring $ theory. More precisely we are mainly interested in extending, in a considerable $ \ l \ part of the thesis, the work done by Abulkheir and Birkenmeier and others in ring l \ l theory to the theory of near-ring ([1],[2],[3],[4],[6],[7],[8],[15],[17],[31],[32],[33].) For the required background, $ we kindly refer the reader to [9],[14],[27],[34],[35]. For the basic notions and $ results used in near-rings we mainly relied on [24] and [25]. Less used and $ universally unknown notions and results will be explicitly explained throughout $ this work in their appropriate places. The thesis consists of four chapters, $ \ l \ and the contents of each are briefly presented in what follows. l \ l Chapter 0 lays down the basic definitions, notions and results, some of which are new as far as we know, needed throughout this work. $ In chapter 1 we introduce basic results and examples. Some of these results are generalizations of recent results concerning SRB rings. Some examples are $ introduced to provide us with means of constructing SRB and SRB* left $ near-rings. In section 1 we show that some special ideals in SRB (SRB*) left $ \ l \ near-rings are essential and then we give equivalences between semiprime, l \ l reduced and right non-singular left near-rings. In the last part of this section we

have some results dealing with reduced and biregular SRB and SRB* near-rings. In section 2 we characterize maximal ideals in the class of SRB (SRB*) near-rings