NU Science Journal 2005; 2(1): 33 - 39 Semigroups with Some Conditions which do not Admit a Distributive Near – ring Structure Manoj Siripitukdet Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000 Thailand E- mail: manojs@nu.ac.th ABSTRACT In this paper, we considered semigroups with some conditions and showed that they did not admit a distributive near – ring structure. Keywords: semigroups, distributive near – ring INTRODUCTION A system ( ) , , S + ⋅ is called a (right) near – ring (Pilz, 1983) if (i) ( ) , S + is a group, (ii) ( ) , S ⋅ is a semigroup and (iii) ( ) x y z xz yz + ⋅ = ⋅ + ⋅ for all , , x yz S ∈ (a right distributive law). For a (right) near – ring ( ) , , S +⋅ , an element d S ∈ is called distributive (Pilz, 1983) if ( ) d x y d x d y ⋅ + = ⋅ + ⋅ for all x, y ∈ S. Let D = {d ∈ S | d is distributive}. A (right) near – ring S is called distributive (Pilz, 1983) if S = D. Then, clearly, S is a distributive near – ring if and only if for all d ∈ S, d is distributive. An element a of a semigroup S is a zero if ax xa a = = for all x S ∈ and we denote a by 0. For any semigroup S, let 0 S S = if S has a zero and S contains more than one element, and otherwise, let 0 S be the semigroup with zero 0 adjoined. For a symbol 1 , S we define 1 S S = if S has an identity, otherwise, let { } 1 1 S S = ∪ if S has no identity. A semigroup S is said to admit a ring structure (Satyanarayana, 1981) if there exists some ring R such that 0 S is isomorphic to the semigroup ( ) , R ⋅ where ⋅ is the multiplication of R, or equivalently, there exists an operation + on S 0 such that ( ) 0 , , S +⋅ is ring where ⋅ is the operation on S 0 . A semigroup admitting a distributive near – ring structure is defined similarly. Let SDN := {S | S is a semigroup admitting a distributive near – ring structure}, SR := {S | S is a semigroup admitting a ring structure}. Clearly, SR ⊆ SDN. That is, if S is a semigroup admitting a ring structure, then S admits a distributive near – ring structure. Semigroups admitting a ring structure