# Some Contributions to Boolean like near Rings

• #### Authors

• K Pushpalatha
• . .
2018-09-01
• Boolean near ring, Boolean like ring, Boolean ring.
• 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.

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• ## References

1. [1] A.L.Foster, The theory of Boolean like rings, Trans.Amer.Math .Soc.59(1946),166-187.

[2] Clay, James R., and Lawyer, Donald A., Boolean near â€“ring, Canad.Math, Bull.12(1969), 265-273.

[3] Gunter Pilz., Near-rings, the Theory and its Applications; North-Holland pub. company, 1983.

[4] Hansen.D.J, and Jiang Luh, Boolean near-rings and weak commutativity, J. Aust. Math.soc (series A) 103-107 (1989).

[5] Pushpalatha. K., On Special Boolean near-rings and Boolean â€“like near-rings, thesis awarded 2015, Acharya Nagarjuna University

[6] Thomas W.Hungerford., Algebra, Holt, Rinehart and Winston Inc., Newyork (1974).

[7] Venkateswarlu.K., et al., Boolean like semirings,int.J.Contemp.Math.Sciences,Vol.6, No.13,619-635.