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Finite near-rings with trivial annihilators

Published online by Cambridge University Press:  09 April 2009

George Szeto
George Szeto
Affiliation:
Mathematics Department, Bradley University, Peoria, Illinois 61606, U. S. A.
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In [3] and [4], the near-rings R with no zero divisors are studied. In particular, a near-ring R is a near-field if it has a non-zero right distributive element ([4], Theorem 1.2.). Also, (R, +) is a nilpotent group if not all non-zero elements of R are left identities of R ([3], Theorem 2). The purpose of the present paper is to extend the above results to a class of near-rings with zero divisors; that is, the set of annihilators of an element x in R, T(x) = {g/xg = 0} is either {0} or R. The examples of such near-rings are those R with (R, +) simple groups and those R with no zero divisors as given in [1], [2], [3] and [4]. For this r, we can easily see that R = A∪S where A = {x/T(x) = R} and S = {x/T(x) = {0}}. Then the second part of this paper will give a structural theorem on the semi-group (S, · ), and more properties on R can be derived.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 2 , September 1974 , pp. 194 - 199
Copyright
Copyright © Australian Mathematical Society 1974

References

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