Centralizer of some special type of additive maps in prime rings

Dhara, Basudeb

doi:10.30504/jims.2026.492627.1223

Centralizer of some special type of additive maps in prime rings

Articles in Press

Document Type : Research Article

Author

Basudeb Dhara

Belda

10.30504/jims.2026.492627.1223

Abstract

Let $R$ be a prime ring with char $(R)\neq 2, 3$ and $f(x_{1}, \ldots, x_{n})$ be a non-central multilinear polynomial over $C(=Z(U))$, where $U$ be the Utumi ring of quotients of $R$. Let $d\neq 0$ be a derivation of $R$. Suppose that $F, G$ are two generalized derivations of $R$. Let $A=\{[(F^2+G)(f(x_1,\ldots,x_n)),f(x_1,\ldots,x_n)] : x_1,\ldots,x_n\in R\}$. In the present article, we determine the nature of the maps $F$ and $G$, when $d(A)=0$. Moreover, it is proved that if $A\neq 0$, then $C_R(A)=Z(R)$, where $C_R (A)=\{a\in R : [a,x]=0 ~~~ \forall x\in A\}$ is the centralizer of $A$.

Keywords

Prime ring

derivation

generalized derivation

extended centroid

Subjects