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Canadian Journal of Mathematics, 1969 The following results (9, Exercise 26, p. 10; 1, Theorem 9.2; 8, Theorem III. 1.11) are known.(A) Let R be a ring with more than one element. Then R is a division ring ifand only if for every a ≠0 in R, there exists a unique b in R such that aba = a.(B) Let R be a near-ring which contains a right identity e ≠ 0.
null Steve Ligh
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Matrix Near-rings over Centralizer Near-rings
Algebra Colloquium, 2000In the most widely accepted definition of matrix near-rings [\textit{J. D. P. Meldrum} and \textit{A. P. J. van der Walt}, Arch. Math. 47, 312-319 (1986; Zbl 0611.16025)], there are two obvious ways of linking ideals in the base near-ring to ideals in the matrix near-ring.
Smith, Kirby C., van Wyk, Leon
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Special radicals of near-rings and Γ-near-rings
Periodica Mathematica Hungarica, 1994All near-rings are 0-symmetric and right distributive. A \(\Gamma\)-near- ring \((M, +, \Gamma)\) is a set \(M\) and a set of binary operators \(\Gamma\) on \(M\) such that \((M, +, \gamma)\) is a near-ring for each \(\gamma \in \Gamma\), and a generalized associative law holds.
Booth, G. L., Veldsman, S.
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On derivations in near-rings and rings
Mathematical Journal of Okayama University, 1992In a near-ring \(R\) a derivation \(D\) is called an scp-derivation if \([x,y] = [D(x), D(y)]\), a Daif 1(2)-derivation if \(D(xy) - D(yx) = [x, y] (=[-x, y])\) (\(\forall x, y \in R\)). Various commutativity (and distributivity) results linked to such derivations are given: e.g.
Bell, Howard E., Mason, Gordon
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Applicable Algebra in Engineering, Communication and Computing, 2020
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Bijan Davvaz +4 more
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Bijan Davvaz +4 more
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Acta Mathematica Hungarica, 1997
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Groenewald, N. J., Olivier, W. A.
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Groenewald, N. J., Olivier, W. A.
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Archiv der Mathematik, 1986
Until this article, there has not been an acceptable approach to the concept of a near-ring of matrices over an arbitrary near-ring. The authors overcome the inherent problems associated with arrays and are motivated by the fact that for a ring, each matrix represents an endomorphism of \((R^ n,+)\) and as such it is derived from the endomorphisms of ...
Meldrum, J. D. P. +1 more
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Until this article, there has not been an acceptable approach to the concept of a near-ring of matrices over an arbitrary near-ring. The authors overcome the inherent problems associated with arrays and are motivated by the fact that for a ring, each matrix represents an endomorphism of \((R^ n,+)\) and as such it is derived from the endomorphisms of ...
Meldrum, J. D. P. +1 more
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Canadian Mathematical Bulletin, 1968
Blackett [4] introduced the concepts of near-ring homomorphism and near-ring ideal. Beidleman [1] established the fundamental homomorphism theorem and the isomorphism theorems for (left) near - rings obeying the condition that 0.a = 0 for every a in the near-ring. Several others, for example [3], [5], and [7], have taken up the study of ideals.
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Blackett [4] introduced the concepts of near-ring homomorphism and near-ring ideal. Beidleman [1] established the fundamental homomorphism theorem and the isomorphism theorems for (left) near - rings obeying the condition that 0.a = 0 for every a in the near-ring. Several others, for example [3], [5], and [7], have taken up the study of ideals.
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New Mathematics and Natural Computation, 2019
In this paper, we consider the problem of how to define [Formula: see text]-nearness ring in the sense of Nobusawa theory which extends the notion of a nearness ring and [Formula: see text]-rings [N. Nobusawa, Osaka J. Math. 1 (1964) 81–89; M. A. Öztürk and E. İnan, Annals of Fuzzy Mathematics and Informatics 17(2) (2019) 115–131].
Mehmet Ali Öztürk, Young Bae Jun
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In this paper, we consider the problem of how to define [Formula: see text]-nearness ring in the sense of Nobusawa theory which extends the notion of a nearness ring and [Formula: see text]-rings [N. Nobusawa, Osaka J. Math. 1 (1964) 81–89; M. A. Öztürk and E. İnan, Annals of Fuzzy Mathematics and Informatics 17(2) (2019) 115–131].
Mehmet Ali Öztürk, Young Bae Jun
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Archiv der Mathematik, 2002
Let \(R\) be a left near-ring and let \(E(R^+)\) be the monoid of endomorphisms of \((R,+)\). For each \(a\in R\), let \(a_\ell\colon R\to R\) be the left-multiplication map \(x\mapsto ax\) and note that \(a_\ell\in E(R^+)\). Let \(L\colon R\to E(R^+)\) be defined by \(L(a)=a_\ell\), and call \(R\) an \(E\)-near-ring if \(L\) is a bijection.
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Let \(R\) be a left near-ring and let \(E(R^+)\) be the monoid of endomorphisms of \((R,+)\). For each \(a\in R\), let \(a_\ell\colon R\to R\) be the left-multiplication map \(x\mapsto ax\) and note that \(a_\ell\in E(R^+)\). Let \(L\colon R\to E(R^+)\) be defined by \(L(a)=a_\ell\), and call \(R\) an \(E\)-near-ring if \(L\) is a bijection.
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