• East Asian mathematical journal
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• v.36 no.1
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• pp.101-114
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• 2020

In this paper, we investigate commutativity of semiprime rings with a derivation which is strongly commutativity preserving and acts as a homomorphism or as an anti-homomorphism on a nonzero Lie ideal.

• Journal of the Chungcheong Mathematical Society
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• v.33 no.3
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• pp.307-318
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• 2020

In this paper, we extend the notion of a generalized derivation F associated with derivation d to two generalized derivations F and G associated with the same derivation d, as a new idea, to obtain the commutativity of prime rings under certain conditions.

• Journal of the Chungcheong Mathematical Society
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• v.29 no.4
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• pp.523-530
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• 2016

Let R be a 3!-torsion free semiprime ring, and let $D:R{\rightarrow}R$ be a Jordan derivation on a semiprime ring R. In this case, we show that [D(x), x]D(x) = 0 if and only if D(x)[D(x), x] = 0 for every $x{\in}R$. In particular, let A be a Banach algebra with rad(A). If D is a continuous linear Jordan derivation on A, then we see that $[D(x),x]D(x){\in}rad(A)$ if and only if $[D(x),x]D(x){\in}rad(A)$ for all $x{\in}A$.

• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.885-898
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• 2012

Let m, n, r be nonzero fixed positive integers, R a 2-torsion free prime ring, Q its right Martindale quotient ring, and L a non-central Lie ideal of R. Let D : $R{\rightarrow}R$ be a skew derivation of R and $E(x)=D(x^{m+n+r})-D(x^m)x^{n+r}-x^mD(x^n)x^r-x^{m+n}D(x^r)$. We prove that if $E(x)=0$ for all $x{\in}L$, then D is a usual derivation of R or R satisfies $s_4(x_1,{\ldots},x_4)$, the standard identity of degree 4.

• East Asian mathematical journal
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• v.16 no.1
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• pp.89-96
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• 2000

In this paper, a differential polynomial ring $R[x;\delta]$ of ring R with a derivation $\delta$ are investigated as follows: For a reduced ring R, a ring R is Baer(resp. quasi-Baer, p.q.-Baer, p.p.-ring) if and only if the differential polynomial ring $R[x;\delta]$ is Baer(resp. quasi-Baer, p.q.-Baer, p.p.-ring).

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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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.

• Kyungpook Mathematical Journal
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• v.59 no.1
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• pp.1-11
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• 2019

Let R be a prime ring with involution of the second kind and centre Z(R). Suppose R admits a generalized derivation $F:R{\rightarrow}R$ associated with a derivation $d:R{\rightarrow}R$. The purpose of this paper is to study the commutativity of a prime ring R satisfying any one of the following identities: (i) $F(x){\circ}x^*{\in}Z(R)$ (ii) $F([x,x^*]){\pm}x{\circ}x^*{\in}Z(R)$ (iii) $F(x{\circ}x^*){\pm}[x,x^*]{\in}Z(R)$ (iv) $F(x){\circ}d(x^*){\pm}x{\circ}x^*{\in}Z(R)$ (v) $[F(x),d(x^*)]{\pm}x{\circ}x^*{\in}Z(R)$ (vi) $F(x){\pm}x{\circ}x^*{\in}Z(R)$ (vii) $F(x){\pm}[x,x^*]{\in}Z(R)$ (viii) $[F(x),x^*]{\mp}F(x){\circ}x^*{\in}Z(R)$ (ix) $F(x{\circ}x^*){\in}Z(R)$ for all $x{\in}R$.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.1
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• pp.1-12
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• 2018

Let R be a 3!-torsion free noncommutative semiprime ring, and suppose there exists a Jordan derivation $D:R{\rightarrow}R$ such that $D^2(x)[D(x),x]=0$ or $[D(x),x]D^2(x)=0$ for all $x{\in}R$. In this case we have $f(x)^5=0$ for all $x{\in}R$. Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D:A{\rightarrow}A$ such that $D^2(x)[D(x),x]{\in}rad(A)$ or $[D(x),x]D^2(x){\in}rad(A)$ for all $x{\in}A$. In this case, we show that $D(A){\subseteq}rad(A)$.

• Communications of the Korean Mathematical Society
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• v.36 no.2
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• pp.229-238
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• 2021

Let R be a ring with characteristic different from 2. An additive mapping F : R → R is called a generalized derivation on R if there exists a derivation d : R → R such that F(xy) = F(x)y + xd(y) holds for all x, y ∈ R. In the present paper, we show that if R is a prime ring satisfying certain identities involving a generalized derivation F associated with a derivation d, then R becomes commutative and in some cases d comes out to be zero (i.e., F becomes a left centralizer). We provide some counter examples to justify that the restrictions imposed in the hypotheses of our theorems are not superfluous.

• Bulletin of the Korean Mathematical Society
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• v.51 no.4
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• pp.1127-1133
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• 2014

Let S be a nonempty subset of a ring R. A map $f:R{\rightarrow}R$ is called strong commutativity preserving on S if [f(x), f(y)] = [x, y] for all $x,y{\in}S$, where the symbol [x, y] denotes xy - yx. Bell and Daif proved that if a derivation D of a semiprime ring R is strong commutativity preserving on a nonzero right ideal ${\rho}$ of R, then ${\rho}{\subseteq}Z$, the center of R. Also they proved that if an endomorphism T of a semiprime ring R is strong commutativity preserving on a nonzero two-sided ideal I of R and not identity on the ideal $I{\cup}T^{-1}(I)$, then R contains a nonzero central ideal. This short note shows that the conclusions of Bell and Daif are also true without the additivity of the derivation D and the endomorphism T.