• Kyungpook Mathematical Journal
• /
• v.55 no.4
• /
• pp.827-835
• /
• 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
• /
• v.33 no.1
• /
• pp.103-125
• /
• 2018

Let R be a 5!-torsion free semiprime ring, and let $D:R{\rightarrow}R$ be a Jordan derivation on a semiprime ring R. Then $[D(x),x]D(x)^2=0$ if and only if $D(x)^2[D(x), x]=0$ for every $x{\in}R$. In particular, let A be a Banach algebra with rad(A) and if D is a continuous linear Jordan derivation on A, then we show that $[D(x),x]D(x)2{\in}rad(A)$ if and only if $D(x)^2[D(x),x]{\in}rad(A)$ for all $x{\in}A$ where rad(A) is the Jacobson radical of A.

• Journal of applied mathematics & informatics
• /
• v.39 no.1_2
• /
• pp.73-81
• /
• 2021

Let R be a prime ring, Qr be the right Martindale quotient ring and C be the extended centroid of R. If �� be a nonzero generalized skew derivation of R and f(x1, x2, ⋯, xn) be a multilinear polynomial over C such that (��(f(x1, x2, ⋯, xn)) - f(x1, x2, ⋯, xn)) ∈ C for all x1, x2, ⋯, xn ∈ R, then either f(x1, x2, ⋯, xn) is central valued on R or R satisfies the standard identity s4(x1, x2, x3, x4).

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

• Bulletin of the Korean Mathematical Society
• /
• v.35 no.3
• /
• pp.583-590
• /
• 1998

The purpose of this paper is to prove the following results: Let A be a noncommutative semisimple Banach algebra. (1) Suppose that a linear derivation D : A $\to A$ is such that [D(x),x]x=0 holds for all $x \in A$. Then we have D=0. (2) Suppose that a linear derivation $D:A\to A$ is such that $D(x)x^2 + x^2D(x)=0$ holds for all $x \in A$. Then we have C=0.

• Journal of the Chungcheong Mathematical Society
• /
• v.27 no.1
• /
• pp.65-87
• /
• 2014

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. We show that if there exists a continuous linear Jordan derivation D : A ${\rightarrow}$ A such that [D(x), x]$D(x)^3{\in}$ rad(A) for all $x{\in}A$, then D(A) ${\subseteq}$ rad(A).

• Communications of the Korean Mathematical Society
• /
• v.17 no.4
• /
• pp.607-618
• /
• 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
• /
• v.5 no.2
• /
• pp.539-546
• /
• 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.

• Journal of the Chungcheong Mathematical Society
• /
• v.30 no.3
• /
• pp.313-321
• /
• 2017

In this paper, we introduce the notion of a ${\ast}$-semiderivation on ${\ast}$-rings, and we try to extend some results for derivations of rings or near-rings to a more general case for ${\ast}$-semiderivations of prime ${\ast}$-rings.

• Communications of the Korean Mathematical Society
• /
• v.28 no.3
• /
• pp.535-558
• /
• 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 $D(A){\subseteq}rad(A)$.