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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Ivan Franko National University of Lviv
2023-09-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/354 |
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| author | N. Rehman E. K. Sogutcu H. M. Alnoghashi |
| author_facet | N. Rehman E. K. Sogutcu H. M. Alnoghashi |
| author_sort | N. Rehman |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-1d58e16bf0044e3d81642f5914009f9b |
| institution | Kabale University |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2023-09-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-1d58e16bf0044e3d81642f5914009f9b2025-08-20T03:28:41ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202023-09-01601122710.30970/ms.60.1.12-27354On generalized homoderivations of prime ringsN. Rehman0E. K. Sogutcu1H. M. Alnoghashi2Aligarh Muslim UniversityDepartment of Mathematics, Cumhuriyet University Sivas, TurkeyDepartment of Computer Science, College of Engineering and Information Technology, Amran University Amran, YemenLet $\mathscr{A}$ 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.http://matstud.org.ua/ojs/index.php/matstud/article/view/354prime ringgeneralized homoderivationcommutativity |
| spellingShingle | N. Rehman E. K. Sogutcu H. M. Alnoghashi On generalized homoderivations of prime rings Математичні Студії prime ring generalized homoderivation commutativity |
| title | On generalized homoderivations of prime rings |
| title_full | On generalized homoderivations of prime rings |
| title_fullStr | On generalized homoderivations of prime rings |
| title_full_unstemmed | On generalized homoderivations of prime rings |
| title_short | On generalized homoderivations of prime rings |
| title_sort | on generalized homoderivations of prime rings |
| topic | prime ring generalized homoderivation commutativity |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/354 |
| work_keys_str_mv | AT nrehman ongeneralizedhomoderivationsofprimerings AT eksogutcu ongeneralizedhomoderivationsofprimerings AT hmalnoghashi ongeneralizedhomoderivationsofprimerings |