β 1 Near - Rings

distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we introduce the notion of β 1 near-rings and study some of their properties. We furnish a complete characterization and also a structure theorem for such near-rings. 1 Introduction A right near-ring is a non-empty set N together with two binary operations " + " and ". " such that (1) (N, +) is a group. (2) (N, .) is a semi-group and (3) (n 1 + n 2)n 3 = n 1 n 3 + n 2 n 3 for all n 1 , n 2 , n 3 ∈ N. Throughout this paper N stands for a right near-ring. (N, +, .) with at least two elements and 0 denotes the identity element of the group (N, +) Obviously, 0n = 0 for all n in N. N is said to be zero-symmetric if n0 = 0 for all n in N. As in [2], a subgroup of (M, +) of (N, +) is called an