Radicals and antiradicals in near-rings
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Date
1979
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University of Nottingham
Abstract
This thesis deals with Jacobson type radicals and ideals, which are antiradical in the sense that they are direct sums of minimal left ideals and annihilate the quasi- radical, Q(N). Of central importance are the s-radical denoted by J 5 (N) and an antiradical called the socle-ideal, denoted by Soi(N). In addition to basic definitions and results, Chapter 1 gives a historical account of some aspects of radical and antiradical theories in near- rings. In Chapter 2 we generalise the notion of s-primitivity, first introduced in (10]. Some of the remaining problems from [10) are settled here. We prove, for example, that if N is a near- ring satisfying the descending chain condition for left ideals, (DCCL), then Js(N) is the smallest two- sided ideal which contains Q(N). We conclude this chapter with examples and some of the typical radical- like properties satisfied by Js(N). Soi(N), which is a generalisation of the Laxton- Machin critical ideal (18], is defined in Chapter 3. If N satisfies the descending chain condition for N-subgroups of N+ (DCCN), then Soi(N) is uniquely maximal amongst all ideals whose intersection with Q(N) is zero. In the DCCL case, Js(N) is uniquely minimal amongst all ideals A such that Soi(N/A) = N/A. We also prove that Soi(N) is contained in the crux of N, first defined by S.D. Scott (27). If N has DCCN, then Soi(N) and the crux of N coincide.
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Keywords
Social ideal, Radical, Antiradical, Near-rings, DCCN