# On generalized homoderivations of prime rings

• N. Rehman Aligarh Muslim University
• E. K. Sogutcu Department of Mathematics, Cumhuriyet University Sivas, Turkey
• H. M. Alnoghashi Department of Computer Science, College of Engineering and Information Technology, Amran University Amran, Yemen
Keywords: Prime ring, generalized homoderivation, commutativity

### Abstract

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.

### Author Biographies

N. Rehman, Aligarh Muslim University

Department of Mathematics, Aligarh Muslim University
Aligarh, India

E. K. Sogutcu, Department of Mathematics, Cumhuriyet University Sivas, Turkey

Department of Mathematics, Cumhuriyet University
Sivas, Turkey

H. M. Alnoghashi, Department of Computer Science, College of Engineering and Information Technology, Amran University Amran, Yemen

Department of Computer Science, College of Engineering
and Information Technology, Amran University
Amran, Yemen

### References

E.F. Alharfie, N.M. Muthana, The commutativity of prime rings with homoderivations, Int. J. Adv. Appl. Sci., 5 (2018), no.5, 79–81.

E.F. Alharfie, N.M. Muthana, Homoderivation of prime rings with involution, Bull. Inter. Math. Virtual Inst., 9 (2019), 305–318.

E.F. Alharfie, N.M. Muthana, On homoderivations and commutativity of rings, Bull. Inter. Math. Virtual Inst., 9 (2019), 301–304.

M.M. El-Sofy, Rings with some kinds of mappings, Master’s thesis, Cairo University, Branch of Fayoum, Cairo, Egypt, 2000.

J.H. Mayne, Centralizing mappings of prime rings, Can. Math. Bull, 27 (1984), no.1, 122–126.

A. Melaibari, N. Muthana, A. Al-Kenani, Homoderivations on rings, Gen. Math. Notes, 35 (2016), no.1, 1–8.

M.K.A. Nawas, R.M. Al-Omary, On ideals and commutativity of prime rings with generalized derivations, Eur. J. Pure Appl. Math., 11 (2018), no.1, 79–89.

N. Rehman, H. Alnoghashi, Identities related to homo-derivation on ideal in prime rings, J. Sib. Fed. Univ., Math. Phys., 16 (2023), no.3, 370–384.

N. Rehman, H. Alnoghashi, On Jordan homo-derivation of triangular algebras, Miskolc Math. Notes, 24 (2023), no.1, 403–410. doi: 10.18514/MMN.2023.3845.

N. Rehman, H. Alnoghashi, Jordan homo-derivations on triangular matrix rings, An. Stiint. Univ. Al. I. Cuza Iasi. Mat., 68 (2022), no.1, 203–216.

N. Rehman, M.M. Rahman, A. Abbasi, Homoderivations on ideals of prime and semi prime rings, Aligarh Bull. Math., 38 (2019), no.1-2, 77–87.

Published
2023-09-22
How to Cite
Rehman, N., Sogutcu, E. K., & Alnoghashi, H. M. (2023). On generalized homoderivations of prime rings. Matematychni Studii, 60(1), 12-27. https://doi.org/10.30970/ms.60.1.12-27
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