• Journal of the Korean Mathematical Society
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• v.47 no.3
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• pp.483-494
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• 2010

Let R be a non-commutative prime ring and I a non-zero left ideal of R. Let U be the left Utumi quotient ring of R and C be the center of U and k, m, n, r fixed positive integers. If there exists a generalized derivation g of R such that $[g(x^m)x^n,\;x^r]_k\;=\;0$ for all x $\in$ I, then there exists a $\in$ U such that g(x) = xa for all x $\in$ R except when $R\;{\cong}\;=M_2$(GF(2)) and I[I, I] = 0.

• Bulletin of the Korean Mathematical Society
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• v.57 no.3
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• pp.709-737
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• 2020

Let R be a prime ring of characteristic different from 2. Suppose that F, G, H and T are generalized derivations of R. Let U be the Utumi quotient ring of R and C be the center of U, called the extended centroid of R and let f(x₁, …, x_n) be a non central multilinear polynomial over C. If F(f(r₁, …, r_n))G(f(r₁, …, r_n)) - f(r₁, …, r_n)T(f(r₁, …, r_n)) = H(f(r₁, …, r_n)²) for all r₁, …, r_n ∈ R, then we describe all possible forms of F, G, H and T.

• Bulletin of the Korean Mathematical Society
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• v.48 no.5
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• pp.917-922
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• 2011

Let R be a prime ring, H a generalized derivation of R, L a noncentral Lie ideal of R, and 0 ${\neq}$ a ${\in}$ R. Suppose that $au^sH(u)u^t$ = 0 for all u ${\in}$ L, where s; t ${\geq}$ 0 are fixed integers. Then H = 0 unless satisfies $S_4$, the standard identity in four variables.

• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.599-605
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• 2009

Let R be a prime ring, H a generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that $u^sH(u)u^t$ = 0 for all u $\in$ L, where s $\geq$ 0, t $\geq$ 0 are fixed integers. Then H(x) = 0 for all x $\in$ R unless char R = 2 and R satisfies $S_4$, the standard identity in four variables.

• 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.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.83-93
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• 2019

Let R be a noncommutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R and f($x_1,{\ldots},x_n$) a noncentral multilinear polynomial over C in n noncommuting variables. Denote by f(R) the set of all the evaluations of f($x_1,{\ldots},x_n$) on R. If d is a nonzero derivation of R and G a nonzero generalized derivation of R such that $$d(G(u)u){\in}Z(R)$$ for all $u{\in}f(R)$, then $f(x_1,{\ldots},x_n)^2$ is central-valued on R and there exists $b{\in}U$ such that G(x) = bx for all $x{\in}R$ with $d(b){\in}C$. As an application of this result, we investigate the commutator $[F(u)u,G(v)v]{\in}Z(R)$ for all $u,v{\in}f(R)$, where F and G are two nonzero generalized derivations of R.

• Journal of the Korean Mathematical Society
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• v.46 no.5
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• pp.997-1005
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• 2009

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and n a fixed positive integer. (i) If (d(x)y+xd(y)+d(y)x+$yd(x))^n$ = xy + yx for all x, y $\in$ I, then R is commutative. (ii) If char R $\neq$ = 2 and (d(x)y + xd(y) + d(y)x + $yd(x))^n$ - (xy + yx) is central for all x, y $\in$ I, then R is commutative. We also examine the case where R is a semiprime ring.