On the Regular Elements of Rings in Which the Product of Any Two Zero Divisors Lies in the Galois Subring

Oduor, Owino Maurice; Libendi, Omamo Aggrey; Christopher, Musoga

doi:10.12732/ijpam.v86i1.2

Cited by 3 publications

(3 citation statements)

References 2 publications

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“…Closely related works can also be seen in Osba et al [5] and Oduor, Omamo and Musoga [6]. Furthermore, Abujabal et al [7] considered the structure and commutativity of general near-rings.…”

Section: Introductionmentioning

confidence: 88%

Regular Elements and Von-Neumann Inverses of a Class of Zero Symmetric Local Near-Rings Admitting Frobenius Derivations

Abuga

Ojiema

Kivunge

2023

ARJOM

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Let N be a zero-symmetric local near-ring. An element \({x}\) \(\in\) N is either regular, zero or a zero divisor. In this paper, we construct a class of zero symmetric local near-ring of characteristic pk; k \(\ge\) 3 admitting an identity frobenius derivation, characterize the structures and orders of the set R(N), the regular compartment with an aim of advancing the classication problem of algebraic structures. The number theoretic notions relating the number of regular elements to Euler's phi-function and the arithmetic functions of Galois near-rings are adopted. Using the Fundamental Theorem of nitely generated Abelian groups, the structures of R(N) are proved to be isomorphic to cyclic groups of various orders. The study also extends to the automorphism groups Aut(R(N)) of the regular elements.

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“…Closely related works can also be seen in Osba et al [5] and Oduor, Omamo and Musoga [6]. Furthermore, Abujabal et al [7] considered the structure and commutativity of general near-rings.…”

Section: Introductionmentioning

confidence: 88%

Regular Elements and Von-Neumann Inverses of a Class of Zero Symmetric Local Near-Rings Admitting Frobenius Derivations

Abuga

Ojiema

Kivunge

2023

ARJOM

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“…Their findings did not consider extensions and idealization using maximal submodules of Zn∀n ∈ Z. Closely related works can also be seen in Osba et al [4] and Oduor, Omamo and Musoga [5]. To obtain a classification of algebraic structures, it is imperative to obtain group structures that are isomorphic to those structures, the automorphism groups.…”

Section: Introductionmentioning

confidence: 99%

Automorphism Groups of Regular Elements with Von-Neumann Inverses of Local Near-Rings Admitting Frobenius Derivations

Abuga

Ojiema

Kivunge

2023

ARJOM

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This paper presents the classication of the invariant subgroups of the automorphism groups of the regular elements obtained from nite local near-rings, the appropriate algebraic structure to study non-linear functions on finite groups. Just as rings of matrices operate on vector spaces, near-rings operate on groups. In this paper, we construct classes of zero symmetric local near-ring of characteristic pk; k = 1; 2 ; k \(\ge\) 3 admitting frobenius derivations, characterize the structures of the cyclic groups generated by the regular elements R(N) as well as the structures and the orders of the automorphism groups Aut(R(N)) of the regular elements.

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“…Unless otherwise stated, J(R) shall denote the Jacobson radical of a completely primary finite ring R. The set of all the regular elements in R shall be denoted by V (R) . The rest of the notations used in this article are standard and reference may be made to [1], [2], [4] and [6].…”

Section: Introductionmentioning

confidence: 99%

On the regular elements of a class of commutative completely primary finite rings

Oduor¹,

Christopher²

2015

ija

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On the Regular Elements of Rings in Which the Product of Any Two Zero Divisors Lies in the Galois Subring

Abstract: Suppose R is a completely primary finite ring in which the product of any two zero divisors lies in the Galois (coefficient) subring. We construct R and find a generalized characterization of its regular elements.

Cited by 3 publications

References 2 publications

Regular Elements and Von-Neumann Inverses of a Class of Zero Symmetric Local Near-Rings Admitting Frobenius Derivations

Regular Elements and Von-Neumann Inverses of a Class of Zero Symmetric Local Near-Rings Admitting Frobenius Derivations

Automorphism Groups of Regular Elements with Von-Neumann Inverses of Local Near-Rings Admitting Frobenius Derivations

On the regular elements of a class of commutative completely primary finite rings

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