• 대한수학회논문집
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• 제33권4호
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• pp.1113-1121
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• 2018

Let R be a prime ring (or semiprime ring) with center Z(R), I a nonzero ideal of R, T an automorphism of $R,S:R^n{\rightarrow}R$ be a symmetric skew n-derivation associated with the automorphism T and ${\Delta}$ is the trace of S. In this paper, we shall prove that S($x_1,{\ldots},x_n$) = 0 for all $x_1,{\ldots},x_n{\in}R$ if any one of the following holds: i) ${\Delta}(x)=0$, ii) [${\Delta}(x),T(x)]=0$ for all $x{\in}I$. Moreover, we prove that if $[{\Delta}(x),T(x)]{\in}Z(R)$ for all $x{\in}I$, then R is a commutative ring.

• Kyungpook Mathematical Journal
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• 제60권4호
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• pp.683-710
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• 2020

Let ℕ be the set of nonnegative integers and 𝕬 be a 2-torsion free triangular algebra over a commutative ring ℛ. In the present paper, under some lenient assumptions on 𝕬, it is proved that if Δ = {𝛿_n}_n∈ℕ is a sequence of ℛ-linear mappings 𝛿_n : 𝕬 → 𝕬 satisfying ${\delta}_n([[x,\;y],\;z])\;=\;\displaystyle\sum_{i+j+k=n}\;[[{\delta}_i(x),\;{\delta}_j(y)],\;{\delta}_k(z)]$ for all x, y, z ∈ 𝕬 with xy = 0 (resp. xy = p, where p is a nontrivial idempotent of 𝕬), then for each n ∈ ℕ, 𝛿_n = d_n + 𝜏_n; where d_n : 𝕬 → 𝕬 is ℛ-linear mapping satisfying $d_n(xy)\;=\;\displaystyle\sum_{i+j=n}\;d_i(x)d_j(y)$ for all x, y ∈ 𝕬, i.e. 𝒟 = {d_n}_n∈ℕ is a higher derivation on 𝕬 and 𝜏_n : 𝕬 → Z(𝕬) (where Z(𝕬) is the center of 𝕬) is an ℛ-linear map vanishing at every second commutator [[x, y], z] with xy = 0 (resp. xy = p).

• 대한수학회보
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• 제49권4호
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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.

• Kyungpook Mathematical Journal
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• 제45권2호
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• pp.249-254
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• 2005

Let N be a prime left near-ring with multiplicative center Z and f be a generalized $({\sigma},{\tau})-derivation$ associated with d. We prove commutativity theorems in prime near- rings with generalized $({\sigma},{\tau})-derivation$.

• Journal of applied mathematics & informatics
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• 제6권2호
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• pp.611-630
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• 1999

In this paper we show that the Derivation D(A) on the non-commutative Banach algebra A with identity satisfying certain conditions is contained in the radical of A and will show some examples satisfying such properties.

• 대한수학회논문집
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• 제34권1호
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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.

• 대한전자공학회:학술대회논문집
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• 대한전자공학회 1999년도 하계종합학술대회 논문집
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• pp.745-748
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• 1999

This paper propose the method to derive RM(Reed-Muller) expansion coefficients for Multiple-Valued function. The general method to obtain RM expansion coefficient for p valued n variable is derivation of single variable transform matrix and expand it n times using Kronecker product. The transform matrix used is p$^{n}$ $\times$ p$^{n}$ matrix. In this case the size of matrix increases depending on the augmentation of variables and the operation is complicated. Thus, to solving the problem, we propose derivation of RM expansion coefficients using p$\times$p transform matrix and Karnaugh-map.

• 대한수학회보
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• 제39권2호
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• pp.211-224
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• 2002

In this paper we show that if D is a continuous linear Jordan derivation on a Banach algebra A satisfying [[D($x^{n}$), $x^{n}$, $x^{n}$] $\in$ rad(A) for a positive integer n and for all x${\in}$A, then D maps A into rad(A).

• Journal of applied mathematics & informatics
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• 제9권1호
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• pp.381-387
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• 2002

Our main goal is to show that if there Jordan derivation D, G on a noncommutative (n+1)!-torsion free prime ring R such that D($\chi$)$\chi$$^n$+$\chi$$^n$G($\chi$) $\in$ C(R) for all $\chi$ $\in$ R, then we have D=0 and G=0.

• Journal of applied mathematics & informatics
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• 제7권1호
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• pp.305-317
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• 2000

It is well-known that every derivation on a commutative Banach algebra maps into its radical. In this paper we shall give the various algebraic conditions on the ring that every Jordan derivation on a noncommutative ring with suitable characteristic conditions is zero and using this result, we show that every continuous linear Jordan derivation on a noncommutative Banach algebra maps into its radical under the suitable conditions.