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Keywords:
prime ring; involution; generalized $(m, n)$-Jordan $*$-centralizer
prime ring; involution; generalized $(m, n)$-Jordan $*$-centralizer
Summary:
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).
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