On generalized homoderivations of prime rings
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{...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | deu |
| Published: |
Ivan Franko National University of Lviv
2023-09-01
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| Series: | Математичні Студії |
| Subjects: | |
| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/354 |
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| Summary: | 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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| ISSN: | 1027-4634 2411-0620 |