Radicals and antiradicals in near-rings
| dc.contributor.author | Hartney, John F.T. | |
| dc.date.accessioned | 2025-08-18T09:17:48Z | |
| dc.date.available | 2025-08-18T09:17:48Z | |
| dc.date.issued | 1979 | |
| dc.description.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. | |
| dc.identifier.uri | https://hdl.handle.net/10566/20735 | |
| dc.language.iso | en | |
| dc.publisher | University of Nottingham | |
| dc.subject | Social ideal | |
| dc.subject | Radical | |
| dc.subject | Antiradical | |
| dc.subject | Near-rings | |
| dc.subject | DCCN | |
| dc.title | Radicals and antiradicals in near-rings | |
| dc.type | Thesis |