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Multiplicative (generalized) (\alpha, \beta)-derivations on left ideals in prime rings
Faiza Shujat 한국전산응용수학회 2022 Journal of Applied and Pure Mathematics Vol.4 No.1
A mapping T:R\to R (not necessarily additive) is called multiplicative left \alpha-centralizer if T(xy)=T(x)\alpha(y) for all x,y\in R. A mapping F:R\to R (not necessarily additive) is called multiplicative (generalized) (\alpha, \beta)-derivation if there exists a map (neither necessarily additive nor derivation) f:R\to R such that F(xy)=F(x)\alpha(y)+\beta(x)f(y) for all x,y\in R, where \alpha and \beta are automorphisms on R. The main purpose of this paper is to study some algebraic identities with multiplicative (generalized) (\alpha, \beta)-derivations and multiplicative left \alpha-centralizer on the left ideal of a prime ring R.
AN IDENTITY ON STANDARD OPERATOR ALGEBRA
SHUJAT, FAIZA The Korean Society for Computational and Applied M 2022 Journal of applied mathematics & informatics Vol.40 No.5-6
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(A<sup>n</sup>B<sup>n</sup>) = D(A<sup>n</sup>)B<sup>n</sup> + A<sup>n</sup>D(B<sup>n</sup>) 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.
MULTIPLICATIVE (GENERALIZED) (𝛼, 𝛽)-DERIVATIONS ON LEFT IDEALS IN PRIME RINGS
SHUJAT, FAIZA The Korean Society for Computational and Applied M 2022 Journal of applied and pure mathematics Vol.4 No.1/2
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.
Semiprime rings with involution and centralizers
Abu Zaid Ansari,Faiza Shujat 한국전산응용수학회 2022 Journal of applied mathematics & informatics Vol.40 No.3
The objective of this research is to prove that an additive mapping T:R\to R is a left as well as right centralizer on R if it satisfies any one of the following identities: \begin{enumerate} \item [$(i)$] $T(x^{n}y^{n}+y^{n}x^{n})=T(x^n)y^{n}+y^nT(x^{n})$ \item [$(ii)$] $2T(x^{n}y^{n})=T(x^n)y^{n}+y^nT(x^{n})$ \end{enumerate} for each x,y\in R, where n\geq 1 is a fixed integer and $R$ is any n!-torsion free semiprime ring. In addition, we talk over above identities in the setting of \ast-ring(ring with involution).
Endomorphisms, anti-endomorphisms and bi-semiderivations on rings
Abu Zaid Ansari,Faiza Shujat,Ahlam Fallatah 한국전산응용수학회 2024 Journal of applied mathematics & informatics Vol.42 No.1
The goal of this study is to bring out the following conclusion: Let $\mathcal{R}$ be a non-commutative prime ring of characteristic not two and $\mathcal{D}$ be a bi-semiderivation on $\mathcal{R}$ with a function $\mathfrak{f}$ (surjective). If $\mathcal{D}$ acts as an endomorphism or as an anti-endomorphism, then $\mathcal{D} =0$ on $\mathcal{R}$.