• Journal of the Chungcheong Mathematical Society
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• v.8 no.1
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• pp.37-44
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• 1995

In this paper we show that every left derivation on a semiprime Banach algebra A is a derivation which maps A into the intersection of the center of A and the Jacobson radical of A, and hence every left derivation on a semisimple Banach algebra is zero.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.659-667
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• 1998

In this paper we show that every left derivation on a semiprime Banach algebra A is a derivation which maps A into the intersection of the center of A and the Jacobson radical of A, and hence every left derivation on a semisimple Banach algebra is always zero.

• Journal of applied mathematics & informatics
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• v.4 no.1
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• pp.263-271
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• 1997

In this paper we show that every left derivation on a semiprime Banach algebra A is a derivation which maps A into the intersection of the center of A and the jacobson radical of A and hence every left derivation on a semisimple Banach algebra is always zero.

• Bulletin of the Korean Mathematical Society
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• v.47 no.1
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• pp.151-157
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• 2010

In this note, we obtain range inclusion results for left Jordan derivations on Banach algebras: (i) Let $\delta$ be a spectrally bounded left Jordan derivation on a Banach algebra A. Then $\delta$ maps A into its Jacobson radical. (ii) Let $\delta$ be a left Jordan derivation on a unital Banach algebra A with the condition sup{r$(c^{-1}\delta(c))$ : c $\in$ A invertible} < $\infty$. Then $\delta$ maps A into its Jacobson radical. Moreover, we give an exact answer to the conjecture raised by Ashraf and Ali in [2, p. 260]: every generalized left Jordan derivation on 2-torsion free semiprime rings is a generalized left derivation.

• Bulletin of the Korean Mathematical Society
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• v.45 no.2
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• pp.253-261
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• 2008

In this paper, we introduce the notion of generalized left derivation on a ring R and prow that every generalized Jordan left derivation on a 2-torsion free primp ring is a generalized left derivation on R. Some related results are also obtained.

• Communications of the Korean Mathematical Society
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• v.17 no.2
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• pp.245-252
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• 2002

In this paper we obtain some results concerning Jordan derivations and Jordan left derivations mapping into the Jacobson radical. Our main result is the following : Let d be a Jordan derivation (resp. Jordan left derivation) of a complex Banach algebra A. If d$^2$(x) = 0 for all x $\in$ A, then we have d(A) ⊆ red(A)

• The Pure and Applied Mathematics
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• v.23 no.4
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• pp.385-387
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• 2016

$Fo{\check{s}}ner$ [4] defined a generalized module left (m, n)-derivation and proved the Hyers-Ulam stability of generalized module left (m, n)-derivations. In this note, we prove that every generalized module left (m, n)-derivation is trival if the algebra is unital and $m{\neq}n$.

• The Pure and Applied Mathematics
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• v.24 no.1
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• pp.33-34
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• 2017

$Fo{\check{s}}ner$ [1] defined a module left (m, n)-derivation and proved the Hyers-Ulam stability of module left (m, n)-derivations. In this note, we prove that every module left (m, n)-derivation is trival if the algebra is unital and $m{\neq}n$.

• Honam Mathematical Journal
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• v.34 no.1
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• pp.55-62
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• 2012

In this article, we take account of stability for ring Jordan left derivations and ring left derivations and we also deal with problems for the radical ranges of linear Jordan left derivations and linear left derivations.

• The Pure and Applied Mathematics
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• v.17 no.4
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• pp.355-362
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• 2010

In this note, we extend the Bresar and Vukman's result [1, Proposition 1.6], which is well-known, to higher left derivations as follows: let R be a ring. (i) Under a certain condition, the existence of a nonzero higher left derivation implies that R is commutative. (ii) if R is semiprime, every higher left derivation on R is a higher derivation which maps R into its center.