Some Contributions to Boolean like near Rings - Pushpalatha, K; ., . (International Journal of Engineering and Technology, 2018)
In this paper we extend Fosterâ€™s Boolean-like ring to Near-rings. We introduce the concept of a Boolean like near-ring.Â A near-ring N is said to be a Boolean-like near-ring if the following conditions hold: (i) a+a = 0 for allÂ Â aÃŽ N , (ii) ab(a+b+ab) = ba for all a, bÂ ÃŽ N and (iii) abc = acb for all a,b, c ÃŽ N (right weak commutative law).Â We have proved that every Boolean ringÂ is a Boolean like near-ring. An example is given to show that the converse is not true.Â We prove thatÂ 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 â€“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. Â Â We show that Â if N is a Boolean-like near-ring with identity, then N is a Boolean-like ring.Â In addition we prove several interesting properties of Â Â Boolean-like near-rings. Â We prove that the set of all nilpotent elements of a Boolean â€“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.Â Â We further prove that every Boolean-like near-ring is a Boolean-like semiring Â Â As example is given to show that the converse of this result is not true. Â Â