Characterizations of Nonadditive Mappings in Prime $*$-Rings Involving Bi-Skew Products

The paper investigates nonadditive mappings \(\Omega: \Re \to \Re\) on a prime ring \(\Re\) with involution \( * \), characterized by satisfying one of the following conditions: \begin{enumerate} \item \( [\Omega(u), \Omega(v)]_\bullet = [u, v]_\bullet \) for all \( u, v \in \Re \). \item \( [\Omega(u), v]_\bullet = [u, \Omega(v)]_\bullet \) for all \( u, v \in \Re \). \item \( \Omega(u \bullet v) = \Omega(u) \bullet v \) for all \( u, v \in \Re \). \end{enumerate} Furthermore, the paper characterizes generalized bi-skew Jordan derivations within prime \( * \)-rings and examines the implications of these results in the context of various operator algebras.