Some Contributions to Boolean like near Rings

Abstract

In this paper we extend Foster’s Boolean-like ring to Near-rings. We introduce the concept of a Boolean like near-ring.  A near-ring N is said to be a Boolean-like near-ring if the following conditions hold: (i) a+a = 0 for all   aÃŽ N , (ii) ab(a+b+ab) = ba for all a, b  ÃŽ N and (iii) abc = acb for all a,b, c ÃŽ N (right weak commutative law).  We have proved that every Boolean ring  is a Boolean like near-ring. An example is given to show that the converse is not true.  We prove that  if N is a Boolean near-ring then conditions (i) and (ii) of the above definition are equivalent. We also proved that a Boolean near-ring with condition (iii) is a Boolean ring. As a consequence we show that a Boolean –like near-ring N is a Boolean ring if and only if it is a Boolean near-ring. Obviously, every Boolean like ring is a Boolean like near-ring.   We show that  if N is a Boolean-like near-ring with identity, then N is a Boolean-like ring.  In addition we prove several interesting properties of   Boolean-like near-rings.  We prove that the set of all nilpotent elements of a Boolean –like near-ring N forms an ideal and the quotient near-ring N/I is a Boolean ring. Every homomorphic image of a Boolean like near ring is a Boolean like near ring.   We further prove that every Boolean-like near-ring is a Boolean-like semiring   As example is given to show that the converse of this result is not true.