• Kyungpook Mathematical Journal
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• v.55 no.4
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• pp.827-835
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• 2015

Let U be a ${\sigma}$-square closed Lie ideal of a 2-torsion free ${\sigma}$-prime ${\Gamma}$-ring M. Let $d{\neq}1$ be an automorphism of M such that $[u,d(u)]_{\alpha}{\in}Z(M)$ on U, $d{\sigma}={\sigma}d$ on U, and there exists $u_0$ in $Sa_{\sigma}(M)$ with $M{\Gamma}u_0{\subseteq}U$. Then, $U{\subseteq}Z(M)$. By applying this result, we generalize the results of Oukhtite and Salhi respect to ${\Gamma}$-rings. Finally, for a non-zero derivation of a 2-torsion free ${\sigma}$-prime $\Gamma$-ring, we obtain suitable conditions under which the $\Gamma$-ring must be commutative.

• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.679-693
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• 2023

Let 𝓡 be a 𝜎-prime ring with involution 𝜎. The main objective of this paper is to describe the structure of the 𝜎-prime ring 𝓡 with involution 𝜎 satisfying certain differential identities involving three derivations 𝜓₁, 𝜓₂ and 𝜓₃ such that 𝜓₁[t₁, 𝜎(t₁)] + [𝜓₂(t₁), 𝜓₂(𝜎(t₁))] + [𝜓₃(t₁), 𝜎(t₁)] ∈ 𝒥_Z for all t₁ ∈ 𝓡. Further, some other related results have also been discussed.

• Kyungpook Mathematical Journal
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• v.60 no.4
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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).

• Honam Mathematical Journal
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• v.32 no.3
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• pp.467-480
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• 2010

There are various papers on finding all the derivations of a non-associative algebra and an anti-symmetrized algebra (see [2], [3], [4], [5], [6], [10], [13], [15], [16]). We and all the derivations of the growing algebra WN($e^{{\pm}x_1x_2x_3}$, 0, 3)[1] with the set of all right annihilators $T_3$ = $\{id,\;\partial_1,\;\partial_2,\;\partial_3\}$ in the paper. The dimension of $Der_{non}$(WN($e^{{\pm}x_1x_2x_3}$, 0, 3)$_{[1]}$) of the algebra WN($e^{{\pm}x_1x_2x_3}$, 0, 3)$_{[1]}$ is one and every derivation of the algebra WN($e^{{\pm}x_1x_2x_3}$, 0, 3)$_{[1]}$ is outer. We show that there is a class P of purely outer algebras in this work.

• Journal of the Chungcheong Mathematical Society
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• v.7 no.1
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• pp.117-122
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• 1994

In this paper we show that if {$H_n$} is a continuous strong higher derivation of order n on an ultraprime Banach algebra with a constant c, then $c||H_1||^2{\leq}4||H_2||$ and for each $1{\leq}l$ < n $$c^2||H_1||\;||H_{n-l}{\leq}6||H_n||+\frac{3}{2}\sum_{\array{i+j+k=n\\i,j,k{\geq}1}}||H_i||\;||H_j||\;||H_k||+\frac{3}{2}\sum_{\array{i+k=n\\i{\neq}l,\;n-1}}||H_i||\;||H_k|| $$ and for a strong higher derivation {$H_n$} of order n on a prime ring A we also show that if [$H_n$(x),x]=0 for all $x{\in}A$ and for every $n{\geq}1$, then A is commutative or $H_n=0$ for every $n{\geq}1$.

• Journal of applied mathematics & informatics
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• v.40 no.5_6
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• pp.1129-1135
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• 2022

The purpose of this research is to find an extension of the renowned Chernoff theorem on standard operator algebra. Infact, we prove the following result: Let H be a real (or complex) Banach space and 𝓛(H) be the algebra of bounded linear operators on H. Let 𝓐(H) ⊂ 𝓛(H) be a standard operator algebra. Suppose that D : 𝓐(H) → 𝓛(H) is a linear mapping satisfying the relation D(AⁿBⁿ) = D(Aⁿ)Bⁿ + AⁿD(Bⁿ) for all A, B ∈ 𝓐(H). Then D is a linear derivation on 𝓐(H). In particular, D is continuous. We also present the limitations on such identity by an example.

• Journal of Applied and Pure Mathematics
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• v.4 no.1_2
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• pp.1-7
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• 2022

A mapping T : R → R (not necessarily additive) is called multiplicative left 𝛼-centralizer if T(xy) = T(x)𝛼(y) for all x, y ∈ R. A mapping F : R → R (not necessarily additive) is called multiplicative (generalized)(𝛼, 𝛽)-derivation if there exists a map (neither necessarily additive nor derivation) f : R → R such that F(xy) = F(x)𝛼(y) + 𝛽(x)f(y) for all x, y ∈ R, where 𝛼 and 𝛽 are automorphisms on R. The main purpose of this paper is to study some algebraic identities with multiplicative (generalized) (𝛼, 𝛽)-derivations and multiplicative left 𝛼-centralizer on the left ideal of a prime ring R.

• East Asian mathematical journal
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• v.38 no.1
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• pp.1-12
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• 2022

Let R be a ring with an endomorphism σ and a σ-derivation δ. R is called (σ, δ)-Baer (resp. (σ, δ)-quasi-Baer, (σ, δ)-p.q.-Baer, (σ, δ)-p.p.) if the right annihilator of every right (σ, δ)-set (resp., (σ, δ)-ideal, principal (σ, δ)-ideal, (σ, δ)-element) of R is generated by an idempotent of R. In this paper, for a given Ore extension A = R[x; σ, δ] of R, the following properties are investigated: If R is a σ-rigid ring in which σ and δ commute, then (1) R is (σ, δ)-Baer if and only if R is (σ, δ)-quasi-Baer if and only if A is (${\bar{\sigma}},\;{\bar{\delta}}$)-Baer if and only if A is (${\bar{\sigma}},\;{\bar{\delta}}$)-quasi-Baer; (2) R is (σ, δ)-p.p. if and only if R is (σ, δ)-p.q.-Baer if and only if A is (${\bar{\sigma}},\;{\bar{\delta}}$)-p.p. if and only if A is (${\bar{\sigma}},\;{\bar{\delta}}$)-p.q.-Baer.

• Journal of applied mathematics & informatics
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• v.36 no.1_2
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• pp.81-92
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• 2018

In this article we study two Multiplicative (generalized)- derivations ${\mathcal{G}}$ and ${\mathcal{H}}$ that satisfying certain conditions in semiprime rings and tried to find out some information about the associated maps. Moreover, an example is given to demonstrate that the semiprimeness imposed on the hypothesis of the various results is essential.

• Journal of applied mathematics & informatics
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• v.7 no.1
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• pp.319-327
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• 2000

In this paper we investigate the conditions for derivations under which the Singer-Wermer theorem is true for noncommutative Banach algebra A such that either [[D(x),xD(x)] ${\in}$ rad(A) for all $x{\in}$A or $D(x)^2$x+xD(x))$^2$${\in}$rad(A) for all $x{\in}$A, where rad(A) is the Jacobson radical of A, then $D(A){\subseteq}$rad(A).