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 \({\cal T}\,:\,{\cal A} \to {{\cal Q}_{ms}}({\cal A})\) . We prove the following under some inevitable torsion restrictions. (a) If m and n are fixed positive integers such that \(\left( {m + n} \right){\cal T}\left( {{a^2}} \right) = m{\cal T}\left( a \right)a^{\ast} + na{\cal T}\left( a \right)\) for all \(a \in {\cal A}\) and \(\left( {m + n} \right){\cal H}\left( {{a^2}} \right) = m{\cal H}\left( a \right)a^{\ast} + na{\cal T}\left( a \right)\) for all \(a \in {\cal A}\) , then \({\cal H} = 0\) . (b) If \({\cal T}\left( {aba} \right) = a{\cal T}\left( b \right)a^{\ast}\) for all \(a,\,b\,\, \in {\cal A}\) , then \({\cal T} = 0\) . Furthermore, we characterize Jordan left τ-centralizers in semiprime rings admitting an anti-automorphism τ. 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).
