Araştırma Makalesi
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Homoderivations in Prime Rings

Yıl 2023, Sayı: 43, 23 - 34, 30.06.2023
https://doi.org/10.53570/jnt.1258402

Öz

The study consists of two parts. The first part shows that if $h_{1}(x)h_{2}(y)=h_{3}(x)h_{4}(y)$, for all $x,y\in R$, then $ h_{1}=h_{3}$ and $h_{2}=h_{4}$. Here, $h_{1},h_{2},h_{3},$ and $h_{4}$ are zero-power valued non-zero homoderivations of a prime ring $R$. Moreover, this study provide an explanation related to $h_{1}$ and $h_{2}$ satisfying the condition $ah_{1}+h_{2}b=0$. The second part shows that $L\subseteq Z$ if one of the following conditions is satisfied: $i. h(L)=(0)$, $ ii. h(L)\subseteq Z$, $iii. h(xy)=xy$, for all $x,y\in L$, $iv. h(xy)=yx$, for all $x,y\in L$, or $v. h([x,y])=0$, and for all $x,y\in L$. Here, $R$ is a prime ring with a characteristic other than $2$, $h$ is a homoderivation of $R$, and $L$ is a non-zero square closed Lie ideal of $R$.

Kaynakça

  • I. N. Herstein, Rings with Involution, University of Chicago Press, Chicago, 1976.
  • M. M. El Sofy Aly, \emph{Rings with Some Kinds of Mappings}, Master's Thesis Cairo University (2000) Cairo.
  • A. Melaibari, N. Muthana, A. Al-Kenani, \emph{Homoderivations on Rings}, General Mathematics Notes 35 (1) (2016) 1{--}8.
  • E. F. Alharfie, N. M. Mthana, \emph{The Commutativity of Prime Rings with Homoderivations}, International Journal of Advanced and Applied Sciences 5 (5) (2018) 79{--}81.
  • E. F. Alharfie, N. M. Mthana, \emph{On Homoderivations and Commutativity of Rings}, Bulletin of the International Mathematical Virtual Institute 9 (2019) 301{--}304.
  • N. Rehman, M. R. Mozumder, A. Abbasi, \emph{Homoderivations on Ideals of Prime and Semiprime Rings}, The Aligarh Bulletin of Mathematics 38 (1-2) (2019) 77{--}87.
  • A. Al-Kenani, A. Melaibari, N. Muthana, \emph{Homoderivations and Commutativity of $\ast -$Prime Rings}, East-West Journal of Mathematics 17 (2) (2015) 117{--}126.
  • A. Melaibari, N. Muthana, A. Al-Kenani, \emph{Centrally-Extended Homoderivations on Rings}, Gulf Journal of Mathematics 4 (2) (2016) 62{--}70.
  • E. F. Alharfie, N. M. Mthana, \emph{Homoderivation of Prime Rings with Involution}, Bulletin of the International Mathematical Virtual Institute 9 (2019) 305{--}318. E. Gselmann, G. Kiss, \emph{Remarks on the Notion of Homo-Derivations}, Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae Sectio Computatorica 51 (2020) 111--130.
  • M. J. Ateyya, \emph{Homogeneralized $\left(\sigma ,\tau\right)-$Derivations of Associative Rings}, in: A. Tercan, A. Gezer, M. Sar\i (Eds.), Studies on Scientific Developments in Geometry, Algebra, and Applied Mathematics, Ankara, 2022, p. 52.
  • M. M. El-Soufi, A. Ghareeb, \emph{Centrally Extended $\alpha $-Homoderivation on Prime and Semiprime Rings}, Hindawi Journal of Mathematics 2022 (2022) Article ID 2584177 5 pages.
  • M. S. Tammam El-Sayiad, A. Ageeb, A. Ghareeb, \emph{Centralizing $n-$Homoderivations of Semiprime Rings}, Hindawi Journal of Mathematics 2022 (2022) Article ID 1112183 8 pages.
  • A. Boua, E. K. S\"{o}\u{g}\"{u}t\c{c}\"{u}, \emph{Semiprime Rings with Generalized Homoderivations}, Boletim da Sociedade Paranaense de Matematica 41 (2023) 8 pages.
  • J. Bergen, I. N.Herstein, J. W. Kerr, \emph{Lie Ideals and Derivations of Prime Rings}, Journal of Algebra 71 (1) (1981) 259{--}267.
  • M. Bresar, \emph{Centralizing Mappings and Derivations in Prime Rings}, Journal of Algebra 156 (1993) 385--394.
  • E. C. Posner, \emph{Derivations in Prime Rings}, Proceedings of the American Mathematical Society 8 (6) (1957) 1093{--}1100.
  • J. H. Mayne, \emph{Centralizing Mappings of Prime Rings}, Canadian Mathematical Bulletin 27 (1) (1984) 122{--}126.
Yıl 2023, Sayı: 43, 23 - 34, 30.06.2023
https://doi.org/10.53570/jnt.1258402

Öz

Kaynakça

  • I. N. Herstein, Rings with Involution, University of Chicago Press, Chicago, 1976.
  • M. M. El Sofy Aly, \emph{Rings with Some Kinds of Mappings}, Master's Thesis Cairo University (2000) Cairo.
  • A. Melaibari, N. Muthana, A. Al-Kenani, \emph{Homoderivations on Rings}, General Mathematics Notes 35 (1) (2016) 1{--}8.
  • E. F. Alharfie, N. M. Mthana, \emph{The Commutativity of Prime Rings with Homoderivations}, International Journal of Advanced and Applied Sciences 5 (5) (2018) 79{--}81.
  • E. F. Alharfie, N. M. Mthana, \emph{On Homoderivations and Commutativity of Rings}, Bulletin of the International Mathematical Virtual Institute 9 (2019) 301{--}304.
  • N. Rehman, M. R. Mozumder, A. Abbasi, \emph{Homoderivations on Ideals of Prime and Semiprime Rings}, The Aligarh Bulletin of Mathematics 38 (1-2) (2019) 77{--}87.
  • A. Al-Kenani, A. Melaibari, N. Muthana, \emph{Homoderivations and Commutativity of $\ast -$Prime Rings}, East-West Journal of Mathematics 17 (2) (2015) 117{--}126.
  • A. Melaibari, N. Muthana, A. Al-Kenani, \emph{Centrally-Extended Homoderivations on Rings}, Gulf Journal of Mathematics 4 (2) (2016) 62{--}70.
  • E. F. Alharfie, N. M. Mthana, \emph{Homoderivation of Prime Rings with Involution}, Bulletin of the International Mathematical Virtual Institute 9 (2019) 305{--}318. E. Gselmann, G. Kiss, \emph{Remarks on the Notion of Homo-Derivations}, Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae Sectio Computatorica 51 (2020) 111--130.
  • M. J. Ateyya, \emph{Homogeneralized $\left(\sigma ,\tau\right)-$Derivations of Associative Rings}, in: A. Tercan, A. Gezer, M. Sar\i (Eds.), Studies on Scientific Developments in Geometry, Algebra, and Applied Mathematics, Ankara, 2022, p. 52.
  • M. M. El-Soufi, A. Ghareeb, \emph{Centrally Extended $\alpha $-Homoderivation on Prime and Semiprime Rings}, Hindawi Journal of Mathematics 2022 (2022) Article ID 2584177 5 pages.
  • M. S. Tammam El-Sayiad, A. Ageeb, A. Ghareeb, \emph{Centralizing $n-$Homoderivations of Semiprime Rings}, Hindawi Journal of Mathematics 2022 (2022) Article ID 1112183 8 pages.
  • A. Boua, E. K. S\"{o}\u{g}\"{u}t\c{c}\"{u}, \emph{Semiprime Rings with Generalized Homoderivations}, Boletim da Sociedade Paranaense de Matematica 41 (2023) 8 pages.
  • J. Bergen, I. N.Herstein, J. W. Kerr, \emph{Lie Ideals and Derivations of Prime Rings}, Journal of Algebra 71 (1) (1981) 259{--}267.
  • M. Bresar, \emph{Centralizing Mappings and Derivations in Prime Rings}, Journal of Algebra 156 (1993) 385--394.
  • E. C. Posner, \emph{Derivations in Prime Rings}, Proceedings of the American Mathematical Society 8 (6) (1957) 1093{--}1100.
  • J. H. Mayne, \emph{Centralizing Mappings of Prime Rings}, Canadian Mathematical Bulletin 27 (1) (1984) 122{--}126.
Toplam 17 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Araştırma Makalesi
Yazarlar

Ayşe Engin 0000-0002-1626-5498

Neşet Aydın 0000-0002-7193-3399

Yayımlanma Tarihi 30 Haziran 2023
Gönderilme Tarihi 1 Mart 2023
Yayımlandığı Sayı Yıl 2023 Sayı: 43

Kaynak Göster

APA Engin, A., & Aydın, N. (2023). Homoderivations in Prime Rings. Journal of New Theory(43), 23-34. https://doi.org/10.53570/jnt.1258402
AMA Engin A, Aydın N. Homoderivations in Prime Rings. JNT. Haziran 2023;(43):23-34. doi:10.53570/jnt.1258402
Chicago Engin, Ayşe, ve Neşet Aydın. “Homoderivations in Prime Rings”. Journal of New Theory, sy. 43 (Haziran 2023): 23-34. https://doi.org/10.53570/jnt.1258402.
EndNote Engin A, Aydın N (01 Haziran 2023) Homoderivations in Prime Rings. Journal of New Theory 43 23–34.
IEEE A. Engin ve N. Aydın, “Homoderivations in Prime Rings”, JNT, sy. 43, ss. 23–34, Haziran 2023, doi: 10.53570/jnt.1258402.
ISNAD Engin, Ayşe - Aydın, Neşet. “Homoderivations in Prime Rings”. Journal of New Theory 43 (Haziran 2023), 23-34. https://doi.org/10.53570/jnt.1258402.
JAMA Engin A, Aydın N. Homoderivations in Prime Rings. JNT. 2023;:23–34.
MLA Engin, Ayşe ve Neşet Aydın. “Homoderivations in Prime Rings”. Journal of New Theory, sy. 43, 2023, ss. 23-34, doi:10.53570/jnt.1258402.
Vancouver Engin A, Aydın N. Homoderivations in Prime Rings. JNT. 2023(43):23-34.


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