• Kyungpook Mathematical Journal
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• v.33 no.1
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• pp.49-51
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• 1993

• Journal of the Korean Mathematical Society
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• v.28 no.2
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• pp.293-308
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• 1991

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.2107-2117
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• 2017

We study the quasi-commutativity in relation with powers of coefficients of polynomials. In the procedure we introduce the concept of ${\pi}$-quasi-commutative ring as a generalization of quasi-commutative rings. We show first that every ${\pi}$-quasi-commutative ring is Abelian and that a locally finite Abelian ring is ${\pi}$-quasi-commutative. The role of these facts are essential to our study in this note. The structures of various sorts of ${\pi}$-quasi-commutative rings are investigated to answer the questions raised naturally in the process, in relation to the structure of Jacobson and nil radicals.

• Communications of the Korean Mathematical Society
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• v.35 no.3
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• pp.723-731
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• 2020

The purpose of this paper is to study the structure of quotient rings R/P where R is an arbitrary ring and P is a prime ideal of R. Especially, we will establish a relationship between the structure of this class of rings and the behavior of derivations satisfying algebraic identities involving prime ideals. Furthermore, the characteristic of the quotient ring R/P has been determined in some situations.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.51-69
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• 2017

We, in this paper, study the commutativity of skew polynomials at zero as a generalization of an ${\alpha}-rigid$ ring, introducing the concept of strongly skew reversibility. A ring R is be said to be strongly ${\alpha}-skew$ reversible if the skew polynomial ring $R[x;{\alpha}]$ is reversible. We examine some characterizations and extensions of strongly ${\alpha}-skew$ reversible rings in relation with several ring theoretic properties which have roles in ring theory.

• Kyungpook Mathematical Journal
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• v.55 no.4
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• pp.827-835
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• 2015

Let U be a ${\sigma}$-square closed Lie ideal of a 2-torsion free ${\sigma}$-prime ${\Gamma}$-ring M. Let $d{\neq}1$ be an automorphism of M such that $[u,d(u)]_{\alpha}{\in}Z(M)$ on U, $d{\sigma}={\sigma}d$ on U, and there exists $u_0$ in $Sa_{\sigma}(M)$ with $M{\Gamma}u_0{\subseteq}U$. Then, $U{\subseteq}Z(M)$. By applying this result, we generalize the results of Oukhtite and Salhi respect to ${\Gamma}$-rings. Finally, for a non-zero derivation of a 2-torsion free ${\sigma}$-prime $\Gamma$-ring, we obtain suitable conditions under which the $\Gamma$-ring must be commutative.

• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.69-76
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• 2012

Let R be a ring, and ${\alpha}$ be an endomorphism of R. An additive mapping H : R ${\rightarrow}$ R is called a left ${\alpha}$-multiplier (centralizer) if H(xy) = H(x)${\alpha}$(y) holds for all x,y $\in$ R. In this paper, we shall investigate the commutativity of prime and semiprime rings admitting left ${\alpha}$-multiplier satisfying any one of the properties: (i) H([x,y])-[x,y] = 0, (ii) H([x,y])+[x,y] = 0, (iii) $H(x{\circ}y)-x{\circ}y=0$, (iv) $H(x{\circ}y)+x{\circ}y=0$, (v) H(xy) = xy, (vi) H(xy) = yx, (vii) $H(x^2)=x^2$, (viii) $H(x^2)=-x^2$ for all x, y in some appropriate subset of R.

• Communications of the Korean Mathematical Society
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• v.32 no.4
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• pp.811-826
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• 2017

The reversible property of rings was initially introduced by Habeb and plays a role in noncommutative ring theory. In this note we study the reversible ring property on nilpotent elements, introducing the concept of commutativity of nilpotent elements at zero (simply, a CNZ ring) as a generalization of reversible rings. We first find the CNZ property of 2 by 2 full matrix rings over fields, which provides a basis for studying the structure of CNZ rings. We next observe various kinds of CNZ rings including ordinary ring extensions.

• Journal of the Korean Mathematical Society
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• v.56 no.2
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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.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.359-370
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• 2022

The principal aim of this paper is to study the connection between the structure of a quotient ring R/P and the behavior of special additive mappings of R. More precisely, we characterize the commutativity of R/P using derivations (generalized derivations) of R satisfying algebraic identities involving the prime ideal P. Furthermore, we provide examples to show that the various restrictions imposed in the hypothesis of our theorems are not superfluous.