Czechoslovak Mathematical Journal, first online, pp. 1-17
On the characterization of certain additive maps in prime $\ast$-rings
Mohammad Ashraf, Mohammad Aslam Siddeeque, Abbas Hussain Shikeh
Received October 15, 2023. Published online April 9, 2024.
Abstract: Let $\mathcal{A}$ be a noncommutative prime ring equipped with an involution `$*$', and let $\mathcal{Q}_{ms}(\mathcal{A})$ be the maximal symmetric ring of quotients of $\mathcal{A}$. Consider the additive maps $\mathcal{H}$ and $\mathcal{T} \colon\mathcal{A}\to\mathcal{Q}_{ms}(\mathcal{A})$. We prove the following under some inevitable torsion restrictions. (a) If $m$ and $n$ are fixed positive integers such that $(m+n)\mathcal{T}(a^2)=m\mathcal{T}(a)a^*+na\mathcal{T}(a)$ for all $a\in\mathcal{A}$ and $(m+n)\mathcal{H}(a^2)=m\mathcal{H}(a)a^*+na\mathcal{T}(a)$ for all $a\in\mathcal{A}$, then $\mathcal{H}=0$. (b) If $\mathcal{T}(aba)=a\mathcal{T}(b)a^*$ for all $a, b\in\mathcal{A}$, then $\mathcal{T}=0$. Furthermore, we characterize Jordan left $\tau$-centralizers in semiprime rings admitting an anti-automorphism $\tau$. As applications, we find the structure of generalized Jordan $*$-derivations in prime rings and generalize as well as improve all the results of A. Abbasi, C. Abdioglu, S. Ali, M. R. Mozumder (2022).
Keywords: prime ring; involution; generalized $(m, n)$-Jordan $*$-centralizer
