Jordan homo-derivations on triangular matrix rings

Rehman, Nadeem ur; Alnoghashi, Hafedh M.

doi:10.47743/anstim.2022.00016

Cited by 1 publication

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“…Again, replacing a by z and b by a in (1), we get ψ(z)ψ(a) − za ∈ Z (A ). Comparing the above expression with (10), we obtain…”

Section: The Main Resultsmentioning

confidence: 99%

“…El Sofy [4], Melaibari et al [6], Alharfie et al [1][2][3], and Rehman et al [11] had shown that if A is a prime ring and ξ : A → A is a homoderivation of A , then A is a commutative ring if A satisfies certain algebraic identities. A growing body of research has been done on homoderivation mappings in rings under different conditions; for example, see [8][9][10].…”

Section: Introduction Throughout a Denotes A Ring With The Center Z (A )mentioning

confidence: 99%

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On generalized homoderivations of prime rings

Rehman,

Sogutcu,

Alnoghashi

2023

Mat. Stud.

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Let $\mathscr{A}$ be a ring with its center $\mathscr{Z}(\mathscr{A}).$ An additive mapping $\xi\colon \mathscr{A}\to \mathscr{A}$ is called a homoderivation on $\mathscr{A}$ if $\forall\ a,b\in \mathscr{A}\colon\quad \xi(ab)=\xi(a)\xi(b)+\xi(a)b+a\xi(b).$ An additive map $\psi\colon \mathscr{A}\to \mathscr{A}$ is called a generalized homoderivation with associated homoderivation $\xi$ on $\mathscr{A}$ if $\forall\ a,b\in \mathscr{A}\colon\quad\psi(ab)=\psi(a)\psi(b)+\psi(a)b+a\xi(b).$ This study examines whether a prime ring $\mathscr{A}$ with a generalized homoderivation $\psi$ that fulfils specific algebraic identities is commutative. Precisely, we discuss the following identities: $\psi(a)\psi(b)+ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(a)\psi(b)-ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(a)\psi(b)+ab\in \mathscr{Z}(\mathscr{A}),$ $\psi(a)\psi(b)-ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(ab)+ab\in \mathscr{Z}(\mathscr{A}),\quad\psi(ab)-ab\in \mathscr{Z}(\mathscr{A}),$ $\psi(ab)+ba\in \mathscr{Z}(\mathscr{A}),\quad\psi(ab)-ba\in \mathscr{Z}(\mathscr{A})\quad (\forall\ a, b\in \mathscr{A}).$ Furthermore, examples are given to prove that the restrictions imposed on the hypothesis of the various theorems were not superfluous.

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“…Again, replacing a by z and b by a in (1), we get ψ(z)ψ(a) − za ∈ Z (A ). Comparing the above expression with (10), we obtain…”

Section: The Main Resultsmentioning

confidence: 99%