• 한국수학교육학회지시리즈B:순수및응용수학
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• 제17권2호
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• pp.131-136
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• 2010

Throught this paper, we will investigate some properties of left regular and strongly reduced near-rings. Mason introduced the notion of left regularity and he characterized left regular zero-symmetric unital near-rings. Also, this concept have been studied by several authors. The purpose of this paper is to find some characterizations of the strong reducibility in near-rings, and the strong regularity in near-rings which are closely related with strongly reduced near-rings.

• 대한수학회논문집
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• 제25권3호
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• pp.343-347
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• 2010

In this paper we study the connection between 2-primal rings, (S,1)-rings and related conditions. And we investigate some condition which is the special case of pseudo symmetric. We also study the condition (KJ) which is given by J. Y. Kim and H. L. Jin. We introduce some condition and we prove that our condition is equivalent to the condition (KJ) when it is an (S,1)-ring.

• 호남수학학술지
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• 제37권4호
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• pp.377-386
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• 2015

The symmetric ring property was due to Lambek and provided many useful results in relation with noncommutative ring theory. In this note we consider this property over centers, introducing symmetric-over-center. It is shown that symmetric and symmetric-over-center are independent of each other. The structure of symmetric-over-center ring is studied in relation to various radicals of polynomial rings.

• 대한수학회보
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• 제56권3호
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• pp.659-666
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• 2019

In this paper, we consider a conjecture of I. N. Herstein for local commutators of symmetric elements and unitary elements of division rings. For example, we show that if D is a finite dimensional division ring with involution ${\star}$ and if $a{\in}D^*=D{\setminus}\{0\}$ such that local commutators $axa^{-1}x^{-1}$ at a are radical over the center F of D for every $x{\in}D^*$ with $x^{\star}=x$, then either $a{\in}F$ or ${\dim}_F\;D=4$.

• Kyungpook Mathematical Journal
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• 제61권1호
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• pp.11-21
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• 2021

We study the structure of rings whose factor rings modulo nonzero proper ideals are right duo; such rings are called right FD. We first see that this new ring property is not left-right symmetric. We prove for a non-prime right FD ring R that R is a subdirect product of subdirectly irreducible right FD rings; and that R/N∗(R) is a subdirect product of right duo domains, and R/J(R) is a subdirect product of division rings, where N∗(R) (J(R)) is the prime (Jacobson) radical of R. We study the relation among right FD rings, division rings, commutative rings, right duo rings and simple rings, in relation to matrix rings, polynomial rings and direct products. We prove that if a ring R is right FD and 0 ≠ e2 = e ∈ R then eRe is also right FD, examining that the class of right FD rings is not closed under subrings.

• East Asian mathematical journal
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• 제15권1호
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• pp.105-109
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• 1999

• 대한수학회지
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• 제56권2호
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• pp.289-309
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• 2019

This paper is intended as a discussion of some generalizations of the notion of a reversible ring, which may be obtained by the restriction of the zero commutative property from the whole ring to some of its subsets. By the INCZ property we will mean the commutativity of idempotent elements of a ring with its nilpotent elements at zero, and by ICZ property we will mean the commutativity of idempotent elements of a ring at zero. We will prove that the INCZ property is equivalent to the abelianity even for nonunital rings. Thus the INCZ property implies the ICZ property. Under the assumption on the existence of unit, also the ICZ property implies the INCZ property. As we will see, in the case of nonunital rings, there are a few classes of rings separating the class of INCZ rings from the class of ICZ rings. We will prove that the classes of rings, that will be discussed in this note, are closed under extending to the rings of polynomials and formal power series.

• East Asian mathematical journal
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• 제15권2호
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• pp.165-176
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• 1999

After the derivation was defined in [19] by Posner a lot of researchers studied the derivations in ring theory in different manners such as in [2], [4], [5], ..., etc. Furthermore, many researches followed the definition of the generalized derivation([3], [6], [7], ..., etc.). Finally, Maksa defined a symmetric bi-derivation and many researches have been done in ring theory by using this definition. In this work, defining a symmetric bi-$\alpha$-derivation, we study the mentioned researches above in the light of this new concept.

• 대한수학회보
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• 제44권1호
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• pp.151-156
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• 2007

A connection between weak ${\pi}-regularity$ and the condition every prime ideal is maximal will be investigated. We prove that a certain 2-primal ring R is weakly ${\pi}-regular$ if and only if every prime ideal is maximal. This result extends several known results nontrivially. Moreover a characterization of minimal prime ideals is also considered.

• 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 $a{\in}N$, $cl(D(a))=V(N^*(N)_a)=Supp(a)=SSpec(N){\setminus}int\;V(a)$.