Abstract
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.
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