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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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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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1987
The principal theorem states that a finite non-constant near-ring N is geometric if and only if it is strongly monogenic. This provides the basis for a well-defined representation of the group space on the group \(\{Z\to aZ+b| \quad a,b\in N,\quad a\neq 0\}\) acting on the underlying set of N.
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The principal theorem states that a finite non-constant near-ring N is geometric if and only if it is strongly monogenic. This provides the basis for a well-defined representation of the group space on the group \(\{Z\to aZ+b| \quad a,b\in N,\quad a\neq 0\}\) acting on the underlying set of N.
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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.
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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.
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Planar Near-Rings, Sandwich Near-Rings and Near-Rings with Right Identity
2005We show that every near-ring containing a multiplicative right identity can be described as a centralizer near-ring with sandwich multiplication. Using this result we characterize planar near-rings and near-rings solving the equation xa=c in terms of such centralizer near-rings with sandwich multiplication.
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Centralizer Near-rings, Matrix Near-rings and Cyclic p-Groups
Algebra Colloquium, 2005If G is a finite group and [Formula: see text] is a group of automorphisms of G, then it is known that the matrix near-ring [Formula: see text] is a subnear-ring of the centralizer near-ring [Formula: see text] for every m ≥ 2. Conditions are known under which [Formula: see text] is a proper subnear-ring of [Formula: see text], and if [Formula: see ...
Smith, Kirby C., van Wyk, Leon
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NEW KINDS OF NEAR-RINGS FROM OLD NEAR RINGS
JP Journal of Algebra, Number Theory and Applications, 2018Summary: In this paper, we construct that a new kind of near-ring, that is, \((e, t)\)-near-ring \((R, +,\ast)\) with given addition in \(R\) and new multiplication \(\ast\) which is expressed in terms of the original multiplication and addition by defining \(a\ast b\) to be a polynomial in \(a\) and \(b\), from a given near-ring \((R, +, \cdot ...
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Embedding of a Near-Ring into a Near-Ring with Identity
1987It is well known, that an arbitrary near-ring N may be embedded into a near-ring N with identity. Details and references are to be found, e.g., in [3; § 1, section c]. Also, it is very well known, that any ring A is an ideal of a ring A* with identity [2; p. 11].
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Mathematical Journal of Okayama University, 1990
Let \(N\) denote a zero-symmetric left near-ring, \(A\) a nonzero ideal of \(N\), and \(d\) a derivation on \(N\). The author proves several theorems on additive or multiplicative commutativity of \(N\), extending results of the reviewer and \textit{G. Mason} [Near-rings and near-fields, Proc. Conf., Tübingen, F.R.G. 1985, North-Holland Math. Stud. 137,
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Let \(N\) denote a zero-symmetric left near-ring, \(A\) a nonzero ideal of \(N\), and \(d\) a derivation on \(N\). The author proves several theorems on additive or multiplicative commutativity of \(N\), extending results of the reviewer and \textit{G. Mason} [Near-rings and near-fields, Proc. Conf., Tübingen, F.R.G. 1985, North-Holland Math. Stud. 137,
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Near-rings whose laminated near-rings are Boolean
九州大学教養部数学雑誌, 1987In [Proc. Edinb. Math. Soc., II. Ser. 23, 97-102 (1980; Zbl 0415.16028)] \textit{K. D. Magill, jun.} introduced the concept of a laminated near-ring. Let N be an arbitrary near-ring. Each element a in N yields a new near- ring \(N_ a\) whose additive group coincides with that of N and whose multiplication * is defined by \(x*y=xay\) for any two ...
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