• Journal of the Chungcheong Mathematical Society
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• v.30 no.2
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• pp.251-258
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• 2017

The aim of the present paper is to establish some results involving generalized ${\ast}$-derivations in ${\ast}$-rings and investigate the commutativity of prime ${\ast}$-rings admitting generalized ${\ast}$-derivations of R satisfying certain identities and some related results have also been discussed.

• The Pure and Applied Mathematics
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• v.25 no.3
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• pp.181-191
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• 2018

In the present paper, we investigate the action of generalized derivation G associated with a derivation g in a (semi-) prime ring R satisfying $(G([x,y])-[G(x),y])^n=0$ for all x, $y{\in}I$, a nonzero ideal of R, where n is a fixed positive integer. Moreover, we also examine the above identity in Banach algebras.

• Communications of the Korean Mathematical Society
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• v.17 no.4
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• pp.607-618
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• 2002

Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation D : A \longrightarrowA such that D($\chi$)[D($\chi$),$\chi$]D($\chi$) $\in$ rad(A) for all $\chi$ A. In this case, we have D(A) ⊆ rad(A).

• Journal of applied mathematics & informatics
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• v.5 no.2
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• pp.539-546
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• 1998

Let A be a noncommutative Banach algebra. Suppose that a continuos linear Jordan derivation D:A$\longrightarrow$A is such that either $[D^2(\chi),\chi^2]\;or\;(D^2(\chi),\chi]+(D(\chi))^2$ lies in the jacobson radical of A for all $\chi$$\in$A. Then D(A) is contained in the Jacobson radical of A.

• Communications of the Korean Mathematical Society
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• v.28 no.3
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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 $D(A){\subseteq}rad(A)$.

• The Pure and Applied Mathematics
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• v.15 no.3
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• pp.259-296
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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 $D(x)^2$[D(x),x] $\in$ rad(A) or [D(x),x]$D(x)^2$ $\in$ rad(A) for all x $\in$ A. In this case, we have D(A) $\subseteq$ rad(A).

• Journal of the Korean Mathematical Society
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• v.44 no.6
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• pp.1267-1279
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• 2007

For a ring endomorphism ${\alpha}$ and an ${\alpha}-derivation\;{\delta}$ of a ring R, we study relation between the set of annihilators in R and the set of annihilators in nearring $R[x;{\alpha},{\delta}]\;and\;R_0[[x;{\alpha}]]$. Also we extend results of Armendariz on the Baer and p.p. conditions in a polynomial ring to certain analogous annihilator conditions in a nearring of skew polynomials. These results are somewhat surprising since, in contrast to the skew polynomial ring and skew power series case, the nearring of skew polynomials and skew power series have substitution for its "multiplication" operation.

• Communications of the Korean Mathematical Society
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• v.32 no.4
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• pp.827-833
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• 2017

In this paper, we define a set including of all $f_a$ with $a{\in}R$ generalized derivations of R and is denoted by $f_R$. It is proved that (i) the mapping $g:L(R){\rightarrow}f_R$ given by g (a) = f-a for all $a{\in}R$ is a Lie epimorphism with kernel $N_{{\sigma},{\tau}}$ ; (ii) if R is a semiprime ring and ${\sigma}$ is an epimorphism of R, the mapping $h:f_R{\rightarrow}I(R)$ given by $h(f_a)=i_{{\sigma}(-a)}$ is a Lie epimorphism with kernel $l(f_R)$ ; (iii) if $f_R$ is a prime Lie ring and A, B are Lie ideals of R, then $[f_A,f_B]=(0)$ implies that either $f_A=(0)$ or $f_B=(0)$.

• Journal of applied mathematics & informatics
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• v.5 no.3
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• pp.819-826
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• 1998

The purpose of this paper is to prove the following results; (1) Let R be a prime ring of char $(R)\neq 2$ and I a nonzero left ideal of R. The existence of a nonzero symmetric bi-derivation D : $R\timesR\;\longrightarrow\;$ such that d is sew-commuting on I where d is the trace of D forces R to be commutative (2) Let m and n be integers with $m\;\neq\;0.\;or\;n\neq\;0$. Let R be a noncommutative prime ring of char$ (R))\neq \; 2-1\; p_1 \;n_1$ where p is a prime number which is a divisor of m, and I a nonzero two-sided ideal of R. Let $D_1$ ; $R\;\times\;R\;\longrightarrow\;and\;$ $D_2\;:\;R\;\times\;R\;longrightarrow\;R$ be symmetric bi-derivations. Suppose further that there exists a symmetric bi-additive mapping B ; $R\;\times\;R\;\longrightarrow\;and\;$ such that $md_1(\chi)\chi ＋ n\chi d_2(\chi)=f(\chi$) holds for all $\chi$$\in$I, where $d_1 \;and\; d_2$ are the traces of $D_1 \;and\; D_2$ respectively and f is the trace of B. Then we have $D_1=0 \;and\; D_2=0$.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.397-408
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• 2016

Let R be a 2-torsion free prime ring with center Z, U be the Utumi quotient ring, Q be the Martindale quotient ring of R, d be a derivation of R and L be a Lie ideal of R. If $d(uv)^n=d(u)^md(v)^l$ or $d(uv)^n=d(v)^ld(u)^m$ for all $u,v{\in}L$, where m, n, l are xed positive integers, then $L{\subseteq}Z$. We also examine the case when R is a semiprime ring. Finally, as an application we apply our result to the continuous derivations on non-commutative Banach algebras. This result simultaneously generalizes a number of results in the literature.