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P. Dheena ¹, C. Jenila ¹

Affiliations
1 Department of Mathematics, Annamalai University, Annamalainagar - 608 002, IN

Abstract

It has been shown that in a near-semiring (S,+,.) with (S,+) as an inverse semigroup, the near-semiring S is strongly regular if and only if S is regular and reduced. In a near-semiring (S,+,.) with (S,+) as an inverse semigroup, equivalent conditions are obtained such that (S,.) is also an inverse semigroup.

Keywords

Inverse Semigroup, Reduced, Regular, Strongly Regular.

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P. Dheena ¹, K. Karthy ¹

Affiliations
1 Department of Mathematics, Annamalai University, Annamalai Nagar-608002, IN

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 π-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].

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P. Dheena ¹, D. Sivakumar ²

Affiliations
1 Department of Mathematics, Annamalai University, Annamalainagar-608 002, IN
2 Department of Mathematics (DDE), Annamalai University, Annamalainagar-608 002, IN

Abstract

In this paper we introduce the notion of N_c-pwe and strict N_c-pure near-rings. We have shown that a reduced near-ring N with identity is N_c-pure if and only if whenever e < a for some e ∉ N_c and ∈ N implies e = ea = and a - e is an idempotent element. We have also obtained an equivalent condition for a N_c pure near-ring to be strict N_cpure.

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P. Dheena ¹

Affiliations
1 Department of Mathematics, Annamalai! University, Annamalainagar—608 002 Tamil Nadu, IN

Abstract

The results proved by Jat and Choudhary II] for left bipotent near-rings without nonzero nilpotent elements are proved by Lee [2] to left r-potent near-rings without nonzero nilpotent elements.

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P. Dheena ¹, C. Rajeswari ¹

Affiliations
1 Department of Mathematics, Annamalai University, Annamalainagar 608 002, IN

Abstract

Throughout this paper N will represent a zero-symmetric right near-ring, and A a non-zero ideal of N. Let d: x → x' be a derivation on N, i.e. an endomorphism of (N, +) satisfying the product rule

                      (xy)' = xy' + x'y for all x, y in N.

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