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Left self distributive near-rings

Published online by Cambridge University Press:  09 April 2009

Gary Birenmeier  and
Henry Heatherly
Gary Birenmeier
Affiliation:
University of Southwestern LouisianaLafayette, Louisiana 70504, U.S.A.
Henry Heatherly
Affiliation:
University of Southwestern LouisianaLafayette, Louisiana 70504, U.S.A.
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Abstract

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This paper considers (left) near-rings which satisfy the left self distributive (LSD) identity: abc = abac. This is exactly the class of near-rings for which each left multiplication mapping, τa: x → ax, is a near-ring endomorphism. Simple and subdirectly irreducible ones are classified and semidirect sum decompositions into reduced and nilpotent pieces are given. LSD near-rings with restrictive conditions on nilpotent elements or annihilating sets are considered. Type 1 prime (semiprime) ideals in an LSD near-ring are completely prime (semiprime). Further results on prime and maximal ideals are given. Numerous examples are given to illuminate the theory and to illustrate its limitations. Some analogous theory for right self distributive near-rings is given (those satisfying the identity: abc = acbc).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 49 , Issue 2 , October 1990 , pp. 273 - 296
Copyright
Copyright © Australian Mathematical Society 1990

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