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N-series and tame near-rings

Published online by Cambridge University Press:  14 November 2011

C. G. Lyons  and
J.D.P. Meldrum
C. G. Lyons
Affiliation:
Department of Mathematics, James Madison University, Harrisonburg, Virginia 22801, U.S.A.
J.D.P. Meldrum
Affiliation:
Department of Mathematics, University of Edinburgh, Edinburgh, Scotland
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Synopsis

Let N be a zero-symmetric near-ring with identity and let G be an N-group. We consider in this paper nilpotent ideals of N and N-series of G and we seek to link these two ideas by defining characterizing series for nilpotent ideals. These often exist and in most cases a minimal characterizing series exists. Another special N-series is a radical series, that is a shortest N-series with a maximal annihilator. These are linked to appropriate characterizing series. We apply these ideas to obtain characterizing series for the radical of a tame near-ring N, and to show that these exist if either G has both chain conditions on N-ideals or N has the descending chain condition on right ideals. In the latter case this provides a new proof of the nilpotency of the radical of a tame near-ring with DCCR, and an internal method for constructing minimal and maximal characterizing series for the radical.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 86 , Issue 1-2 , 1980 , pp. 153 - 163
Copyright
Copyright © Royal Society of Edinburgh 1980

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