On Regular Laminated Near-Rings

Abstract

Let N be a near-ring and a∈N. A new product * may be defined on N by x*y=xay. It is clear that (N,+, *) is a near-ring called laminated near-ring. Throughout this paper (N,+,*) stands for the laminated near-ring laminated by the element a. Yakabe [5] obtained results on Boolean laminated near-ring. In this paper we have obtained results on regular laminated near-rings. N is said to be regular if given a∈N, there is an x∈N such that a=axa. N is said to be strongly regular if given a∈N, there is an x∈N such that a=xa². N is said to be Ï€-regular if given a∈N, there exist n≥1 and y∈N such that aⁿ=aⁿyaⁿ. N is said to be unit regular if for every x in N there exists a unit u in N such that x=xux. N is said to be strongly clean, if every element in N can be written as a sum of idempotent and an invertible element and they commute. N has stable range one if for any b, c in N satisfying bx+c=1, there exists a y in such that b+cy is a unit in N. Throughout this paper N stands for a zero symmetric right near-ring with identity. For the basic terminology and notation we refer to Pilz [4].