• East Asian mathematical journal
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• 제28권3호
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• pp.281-285
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• 2012

In this paper, the concept of *-prime ideals in non-associative near-rings is introduced and then will be studied. For this purpose, first we introduce the notions of *-operation, *-prime ideal and *-system in a near-ring. Next, we will define the *-sequence, *-strongly nilpotent *-prime radical of near-rings, and then obtain some characterizations of *-prime ideal and *-prime radical $r_s$(I) of an ideal I of near-ring N.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제8권2호
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• pp.145-152
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• 2001

Posner [Proc. Amer. Math. Soc. 8 (1957), 1093-1100] defined a derivation on prime rings and Herstein [Canad, Math. Bull. 21 (1978), 369-370] derived commutative property of prime ring with derivations. Recently, Bergen [Canad. Math. Bull. 26 (1983), 267-227], Bell and Daif [Acta. Math. Hunger. 66 (1995), 337-343] studied derivations in primes and semiprime rings. Also, in near-ring theory, Bell and Mason [Near-Rungs and Near-Fields (pp. 31-35), Proceedings of the conference held at the University of Tubingen, 1985. Noth-Holland, Amsterdam, 1987; Math. J. Okayama Univ. 34 (1992), 135-144] and Cho [Pusan Kyongnam Math. J. 12 (1996), no. 1, 63-69] researched derivations in prime and semiprime near-rings. In this paper, Posner, Bell and Mason＇s results are extended in prime near-rings with some conditions.

• 충청수학회지
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• 제29권4호
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• pp.643-650
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• 2016

In this paper, we introduce the notion of a semiderivation on near-rings, and we try to generalize some properties of prime rings with derivations to prime near-rings with semiderivations.

• 대한수학회지
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• 제58권2호
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• pp.309-326
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• 2021

In near-ring theory several different types of primitivity exist. These all imply several different types of primeness. In case of near-rings with DCCN most of the types of primeness are known to imply primitivity of a certain kind. We are able to show that also so called 1-prime near-rings imply 1-primitivity. This enables us to classify maximal ideals in near-rings with chain condition with the concept of 1-primeness which leads to further results in the structure theory of near-rings.

• 대한수학회논문집
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• 제34권1호
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• pp.17-26
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• 2019

In this paper, we investigate commutativity of 3-prime near-rings ${\mathcal{N}}$ in which (1, ${\alpha}$)-derivations satisfy certain algebraic identities. Some well-known results characterizing commutativity of 3-prime near-rings have been generalized. Furthermore, we give some examples show that the restriction imposed on the hypothesis is not superfluous.

• Kyungpook Mathematical Journal
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• 제45권2호
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• pp.249-254
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• 2005

Let N be a prime left near-ring with multiplicative center Z and f be a generalized $({\sigma},{\tau})-derivation$ associated with d. We prove commutativity theorems in prime near- rings with generalized $({\sigma},{\tau})-derivation$.

• 대한수학회논문집
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• 제31권3호
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• pp.433-445
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• 2016

In the present paper, we expand the domain of work on the concept of semiderivations in 3-prime near-rings through the study of structure and commutativity of near-rings admitting semiderivations satisfying certain differential identities. Moreover, several examples have been provided at places which show that the assumptions in the hypotheses of various theorems are not altogether superfluous.

• 대한수학회논문집
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• 제28권4호
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• pp.669-677
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• 2013

In this paper we introduce the notion of NI near-rings similar to the notion introduced in rings. We give topological properties of collection of strongly prime ideals in NI near-rings. We have shown that if N is a NI and weakly pm near-ring, then $Max(N)$ is a compact Hausdorff space. We have also shown that if N is a NI near-ring, then for every $a{\in}N$, $cl(D(a))=V(N^*(N)_a)=Supp(a)=SSpec(N){\setminus}int\;V(a)$.

• 대한수학회논문집
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• 제33권1호
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• pp.37-44
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• 2018

We prove some theorems showing that a right Jordan ideal or a left Jordan ideal of a 3-prime near-ring must be commutative if it admits a nonzero derivation acting as a homomorphism or an antihomomorphism. Moreover, we give examples proving necessity of the conditions given.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제13권4호
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• pp.303-310
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• 2006

M. $Bre\v{s}ar$ and J. Vukman obtained some results concerning orthogonal derivations in semiprime rings which are related to the result that is well-known to a theorem of Posner for the product of two derivations in prime rings. In this paper, we present orthogonal generalized derivations in semiprime near-rings.