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A radical for near-rings

Published online by Cambridge University Press:  14 November 2011

J. F. T. Hartney
J. F. T. Hartney
Affiliation:
Mathematics Department, University of the Witwatersrand, Johannesburg, South Africa
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Synopsis

Throughout this paper the near-ring N is assumed to be zero symmetric and to satisfy the right distributive law. That is, x · 0 = 0 and (x + y)z = xz + yz for all x, y, z ∈ N. In what follows we generalise the notion of s-primitivity first introduced in an earlier paper by the author (1968), where only distributively generated (d.g.) near-rings with identity were considered. We define a Jacobson type radical Js (N) and show that J1(N)⊇Js(N) ⊇ Q(N), where Q(N) is the intersection of all 0-modular left ideals of N (Pilz). In addition we settle some of the problems remaining from Hartney (1968).

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 93 , Issue 1-2 , 1982 , pp. 105 - 110
Copyright
Copyright © Royal Society of Edinburgh 1982

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References

1Baer, Reinhold. Radical ideals. Amer. J. Math. 65 (1943), 537–568.CrossRefGoogle Scholar
2Betsch, Gerhard. Struktursatze für Fastringe (Inaugural Disertation, Eberhard-Karls-Universität, Tubingen).Google Scholar
3Blackett, D. W.. Simple and Semi-simple near-rings. Proc. Amer. Math. Soc. 4 (1953), 772–785.CrossRefGoogle Scholar
4Hall, J. D.. M. Phil, thesis, University of Nottingham (1973).Google Scholar
5Hartney, J. F. T.. On the radical theory of a distributively generated near-ring. Math. Scand. 23 (1968), 214–220.CrossRefGoogle Scholar
6Hartney, J. F. T.. Ph.D. thesis, University of Nottingham (1979).Google Scholar
7Laxton, R. R.. A radical and its theory for distributively generated near-rings. J. London. Math. Soc. 38 (1963), 40–49.CrossRefGoogle Scholar
8Laxton, R. R.. Prime ideas and the ideal radical of a distributively generated near-ring. Math. Z. 83 (1964), 8–17.CrossRefGoogle Scholar
9Laxton, R. R.. Note on the radical of a near-ring. J. London Math. Soc. 6 (1972), 12–14.CrossRefGoogle Scholar
10Laxton, R. R. and Machin, A. W.. On the decomposition of near-rings. Abh. Math. Sem. Univ. Hamburg 38 (1972), 221–230.CrossRefGoogle Scholar
11Machin, A. W.. Ph.D. thesis, University of Nottingham (1971).Google Scholar
12Neumann, Hanna. On varieties of groups and their associated near-rings. Math. Z. 65 (1956), 36–69.CrossRefGoogle Scholar
13Pilz, Günter. Near-rings. North Holland Math. Studies 23 (1977).Google Scholar
14Ramakotaiah, D.. Radicals for near-rings. Math. Z. 97 (1967), 45–56.CrossRefGoogle Scholar