Structure of some additive maps in prime rings with involution

Let \(\textrm{R}\) be a noncommutative prime ring equipped with an involution ‘\(*\)’, and let \(\mathcal {Q}_{ml}(\textrm{R})\) be the maximal left ring of quotients of \(\textrm{R}\). The objective of this paper is to characterize additive maps \(\mathcal {H}:\textrm{R}\rightarrow \mathcal {Q}_{ml}(\textrm{R})\) that satisfy any one of the following conditions. (i) \(\mathcal {H}(srs)=\mathcal {H}(s)s^*r^*+s\mathcal {H}(r)s^*+sr\mathcal {H}(s)\) for all \(s, r\in \textrm{R}\). (ii) \(\mathcal {H}(s^*s)=\mathcal {H}(s^*)s+s^*\mathcal {H}(s)\) for all \(s\in \textrm{R}\).