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On Regular Laminated Near-Rings


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1 Department of Mathematics, Annamalai University, Annamalai Nagar-608002, India
     

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Let N be a near-ring and aN. 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 aN, there is an xN such that a=axa. N is said to be strongly regular if given aN, there is an xN such that a=xa2. N is said to be π-regular if given a∈N, there exist n≥1 and yN 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].
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  • On Regular Laminated Near-Rings

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Authors

P. Dheena
Department of Mathematics, Annamalai University, Annamalai Nagar-608002, India
K. Karthy
Department of Mathematics, Annamalai University, Annamalai Nagar-608002, India

Abstract


Let N be a near-ring and aN. 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 aN, there is an xN such that a=axa. N is said to be strongly regular if given aN, there is an xN such that a=xa2. N is said to be π-regular if given a∈N, there exist n≥1 and yN 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].