Prior to 1990, the only well known ideal-hereditary Kurosh-Amitsur radicals in the variety of zero-symmetric near-rings were the Jacobson type radicals Iv(N) , where â¨â{2,3} and the Brown-McCoy radical. In 1990, Booth, Groenewald and Veldsman introduced the concept of an equiprime near-ring which leads to an ideal-hereditary Kurosh-Amitsur radical in Nâ. The concept of an equiprime near-ring generalizes the concept of a prime ring to near-rings. Although the search for more ideal-hereditary radicals of near-rings was apparently the original motivation for the introduction of equiprime near-rings, it became clear that these near-rings are interesting in their own right. It is our aim in this treatise to give an exposition of the many interesting properties of equiprime near-rings. We begin with a brief reminder of near-ring rudiments; giving basic definitions and elementary results which are necessary for understanding and development of subsequent chapters. With the basics out of the way, our main task begins with a consideration of equiprime, strongly and completely equiprime left ideals. It is noted that any zero-symmetric near-ring can be embedded in an equiprime near-ring. Moreover, the class of equiprime near-rings is shown to be hereditary. Open questions arising out of the study of equiprime near-rings are highlighted along the way. In Chapter 3 we consider well known examples of near-rings and determine when such near-rings are equiprime. This provides more insight into the nature of equiprime near-rings and is a fertile ground for the birth of examples and counterexamples which may be used to close or solve some open question within the literature. We also prove some results which generalize some results of Booth and Hall [10] and Veldsman [29]. These results have not been previously presented elsewhere to the best of our knowledge. vii In Chapter 4, the equiprime near-rings are shown to yield an ideal-hereditary radical in Nâ. It is shown that a special radical theory can be built on the equiprime nearrings in much the same way prime rings are used in ring theory to define special radical classes of rings.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:nmmu/vital:10505 |
| Date | January 2008 |
| Creators | Mogae, Kabelo |
| Publisher | Nelson Mandela Metropolitan University, Faculty of Science |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis, Masters, MSc |
| Format | vii, 84 leaves, pdf |
| Rights | Nelson Mandela Metropolitan University |
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