TY - JOUR
AU - Rehman, N.
AU - Sogutcu, E. K.
AU - Alnoghashi, H. M.
PY - 2023/09/22
Y2 - 2024/07/12
TI - On generalized homoderivations of prime rings
JF - Matematychni Studii
JA - Mat. Stud.
VL - 60
IS - 1
SE - Articles
DO - 10.30970/ms.60.1.12-27
UR - http://www.matstud.org.ua/ojs/index.php/matstud/article/view/354
SP - 12-27
AB - 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.
ER -