• 대한수학회보
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• 제43권3호
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• pp.635-644
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• 2006

For a lower niltriangular matrix ring $A=NT_n(K)(n{\geq}3)$, we show that every derivation of A is a sum of certain diagonal, trivial extension and strongly nilpotent derivation. Moreover, a strongly nilpotent derivation is a sum of an inner derivation and an uaz-derivation.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제20권4호
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• pp.243-249
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• 2013

Let A be an algebra and D a derivation of A. Then D is called algebraic nil if for any $x{\in}A$ there is a positive integer n = n(x) such that $D^{n(x)}(P(x))=0$, for all $P{\in}\mathbb{C}[X]$ (by convention $D^{n(x)}({\alpha})=0$, for all ${\alpha}{\in}\mathbb{C}$). In this paper, we show that any algebraic nil derivation (possibly unbounded) on a commutative complex algebra A maps into N(A), where N(A) denotes the set of all nilpotent elements of A. As an application, we deduce that any nilpotent derivation on a commutative complex algebra A maps into N(A), Finally, we deduce two noncommutative versions of algebraic nil derivations inclusion range.

• 대한수학회보
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• 제26권1호
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• pp.31-34
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• 1989

In this paper we show that if there is a derivation on a commutative Banach algebra which has a non-nilpotent separating space, then there is a discontinuous derivation on a commutative Banach algebra which has a range in its radical. Also we show that if every prime ideal is closed in a commutative Banach algebra with identity then every derivation on it has a range in its radical.

• Journal of applied mathematics & informatics
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• 제39권5_6호
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• pp.761-768
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• 2021

Let I be a nonzero ideal of a prime ring R and m, n are positive integers. If R admits a generalized derivation F satisfying F(x)ⁿF(y)^m - xⁿy^m = 0 for any y, x ∈ I, then F(x) = ax for any x ∈ R and a^m+n = 1, where a ∈ C, the extended centroid of R.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제21권4호
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• pp.237-246
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• 2014

Let M be a prime ${\Gamma}$-ring and let d be a derivation of M. If there exists a fixed integer n such that $(d(x){\alpha})^nd(x)=0$ for all $x{\in}M$ and ${\alpha}{\in}{\Gamma}$, then we prove that d(x) = 0 for all $x{\in}M$. This result can be extended to semiprime ${\Gamma}$-rings.

• 대한수학회보
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• 제61권3호
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• pp.867-873
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• 2024

A pseudo-Riemannian solvmanifold is a solvable Lie group endowed with a left invariant pseudo-Riemannian metric. In this short note, we investigate the nilpotency of the Ricci operator of pseudo-Riemannian solvmanifolds. We focus on a special class of solvable Lie groups whose Lie algebras can be expressed as a one-dimensional extension of a nilpotent Lie algebra ℝD⋉n, where D is a derivation of n whose restriction to the center of n has at least one real eigenvalue. The main result asserts that every solvable Lie group belonging to this special class admits a left invariant pseudo-Riemannian metric with nilpotent Ricci operator. As an application, we obtain a complete classification of three-dimensional solvable Lie groups which admit a left invariant pseudo-Riemannian metric with nilpotent Ricci operator.

• 대한수학회보
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• 제42권4호
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• pp.671-678
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• 2005

Let R be a noncommutative semi prime ring. Suppose that there exists a derivation d : R $\to$ R such that for all x $\in$ R, either [[d(x),x], d(x)] = 0 or $\langle$$\langle(x),\;x\rangle,\;d(x)\rangle$ = 0. In this case [d(x), x] is nilpotent for all x $\in$ R. We also apply the above results to a Banach algebra theory.

• 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.

• East Asian mathematical journal
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• 제18권2호
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• pp.195-203
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• 2002

Let R be a prime ring with characteristic not two and U is a nonzero left ideal of R which contains no nonzero nilpotent right ideal as a ring. For a $(\sigma,\tau)$-derivation d : R$\rightarrow$R, we prove the following results: (1) If d is an endomorphism on R then d=0. (2) If d is an anti-endomorphism on R then d=0. (3) If d(xy)=d(yx), for all x, y$\in$R then R is commutative. (4) If d is an homomorphism or anti-homomorphism on U then d=0.

• 충청수학회지
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• 제14권1호
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• pp.21-28
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• 2001

In this paper we show that the following results remain valid for arbitrary Jordan derivations as well: Let d be a derivation of a complex Banach algebra A. If $d^2(x){\in}rad(A)$ for all $x{\in}A$, then we have $d(A){\subseteq}rad(A)$ ([5, p. 243]), and in a case when A is unital, $d(A){\subseteq}rad(A)$ if and only if sup{$r(z^{-1}d(z)){\mid}z{\in}A$ invertible} < ${\infty}$([3]), where rad(A) stands for the Jacobson radical of A, and r(${\cdot}$) for the spectral radius.