A note on certain additive maps in prime rings with involution

Abstract

Let ${\mathcal {A}}$ be a noncommutative prime ring equipped with an involution $`*$’. Let ${\mathcal {Q}}_{ms}({\mathcal {A}})$ be the maximal symmetric ring of quotients of ${\mathcal {A}}$ and ${\mathcal {C}}$ be the extended centroid of ${\mathcal {A}}$. The objective of this manuscript is to characterize the additive map ${\mathcal {T}}:{\mathcal {A}}\rightarrow {\mathcal {Q}}_{ms}({\mathcal {A}})$ satisfying any one of the following conditions.

(i):

${\mathcal {T}}(a^2) = a{\mathcal {T}}(a)+\eta {\mathcal {T}}(a)a^*$ for all $a\in {\mathcal {A}}$, where $\eta \in {\mathcal {C}}$ and $\eta \notin \{0, 1\}$.

(ii):

${\mathcal {T}}(a)(a^*)^n+a^n{\mathcal {T}}(a)=0$ for all $a\in {\mathcal {A}}$, where n is a fixed positive integer.

We also study some results concerning generalized derivations in prime rings with involution. Moreover, all the study in the manuscript is carried out in the light of the theory of functional identities.