• Kyungpook Mathematical Journal
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• 제53권4호
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• pp.497-505
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• 2013

The noncommutative Singer-Wermer conjecture states that every derivation on a Banach algebra (possibly noncommutative) leaves primitive ideals of the algebra invariant. This conjecture is still an open question for more than thirty years. In this note, we approach this question via some sufficient conditions for the separating ideal of ${\phi}$-derivations to be nilpotent. Moreover, we show that the spectral boundedness of ${\phi}$-derivations implies that they leave each primitive ideal of Banach algebras invariant.

• 대한수학회논문집
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• 제28권3호
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• pp.535-558
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• 2013

The purpose of this paper is to prove that the noncommutative version of the Singer-Wermer Conjecture is affirmative under certain conditions. Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D:A{\rightarrow}A$ such that $D(x)^3[D(x),x]{\in}rad(A)$ for all $x{\in}A$. In this case, we show that $D(A){\subseteq}rad(A)$.

• 대한수학회보
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• 제30권1호
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• pp.61-64
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• 1993

In this paper, we show that if R is a local-global domain then the Question holds. McDonald and Waterhouse in [6] and Estes and Guralnick in [5] introduced the concept of local-global rings (so called rings with many units) independently. A local-global ring is a commutative ring R with 1 satisfying; if a polynomial f in R[ $x_{1}$, .., $x_{n}$] represents a unit over $R_{P}$ for every maximal ideal P in R, then f represents a unit over R. Such rings include semilocal rings, or more generally, rings which are von Neumann regular mod their Jacobson radical, and the ring of all algebraic integers.s.s.

• 대한수학회지
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• 제49권3호
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• pp.605-622
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• 2012

A ring $R$ is called semiregular if $R/J$ is regular and idem-potents lift modulo $J$, where $J$ denotes the Jacobson radical of $R$. We give some characterizations of rings $R$ such that idempotents lift modulo $J$, and $R/J$ satisfies one of the following conditions: (one-sided) unit-regular, strongly regular, (unit, strongly, weakly) ${\pi}$-regular.

• 대한수학회논문집
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• 제34권3호
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• pp.811-818
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• 2019

Let A be a Banach algebra with rad(A). We show that if there exists a continuous linear Jordan derivation D on A, then $$[D(x),x]D(x)^2{\in}rad(A)$$ if and only if $D(x)[D(x),x]D(x){\in}rad(A)$ for all $x{\in}A$.

• 대한수학회보
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• 제39권2호
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• pp.211-224
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• 2002

In this paper we show that if D is a continuous linear Jordan derivation on a Banach algebra A satisfying [[D($x^{n}$), $x^{n}$, $x^{n}$] $\in$ rad(A) for a positive integer n and for all x${\in}$A, then D maps A into rad(A).

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제16권2호
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• pp.227-241
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• 2009

Let A be a Banach algebra. Suppose there exists a continuous linear Jordan derivation D : A $\rightarrow$ A such that $[D(x),\;x]D(x)^2[D(x),\;x]\;{\in}\;rad(A)$ for all $x\;{\in}\;A$. Then we have D(A) $\subseteq$ rad(A).

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제15권2호
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• pp.179-201
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• 2008

Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D\;:\;A{\rightarrow}A$ such that $D(x)[D(x),x]^2\;{\in}\;rad(A)$ or $[D(x), x]^2 D(x)\;{\in}\;rad(A)$ for all $x\;{\in}\ A$. In this case, we have $D(A)\;{\subseteq}\;rad(A)$.

• Korean Journal of Mathematics
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• 제22권1호
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• pp.151-167
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• 2014

Let M, N be modules over a ring R and $[M,N]=Hom_R(M,N)$. The concern is study of: (1) Some fundamental properties of [M, N] when [M, N] is regular or semipotent. (2) The substructures of [M, N] such as radical, the singular and co-singular ideals, the total and others has raised new questions for research in this area. New results obtained include necessary and sufficient conditions for [M, N] to be regular or semipotent. New substructures of [M, N] are studied and its relationship with the Tot of [M, N]. In this paper we show that, the endomorphism ring of a module M is regular if and only if the module M is semi-injective (projective) and the kernel (image) of every endomorphism is a direct summand.

• 대한수학회지
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• 제53권2호
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• pp.475-488
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• 2016

We study the structure of central elements in relation with polynomial rings and introduce quasi-commutative as a generalization of commutative rings. The Jacobson radical of the polynomial ring over a quasi-commutative ring is shown to coincide with the set of all nilpotent polynomials; and locally finite quasi-commutative rings are shown to be commutative. We also provide several sorts of examples by showing the relations between quasi-commutative rings and other ring properties which have roles in ring theory. We examine next various sorts of ring extensions of quasi-commutative rings.