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Let S be a unital prime ∗-ring containing a nontrivial symmetric idempotent and let m s (S) be the maximal symmetric ring of quotients of S. Using the technique of Peirce decomposition and the theory of functional identities, we prove that a map : S → m s (S) satisfies (u v ∗ + v u ∗) = (u) v ∗ + (v) u ∗ + u (v) ∗ + v (u) ∗ for all u , v ∈ S if and only if is an additive ∗-derivation unless dim S = 4 and char(S) = 2. As an application, we shall characterize such maps in different operator algebras. [ABSTRACT FROM AUTHOR]

Let * be an involution of a non-commutative prime ring ℜ with the maximal symmetric ring of quotients and the extended centroid of ℜ denoted by Q m s (ℜ) and C , respectively. Consider : ℜ → Q m s (ℜ) be an additive map, if (u 2) − (u) u * − u (u) ∈ C for all u ∈ ℜ , then such a map is termed as a weak Jordan *-derivation. With the smart handling of the FI-theory and facing the challenging case of low dimensions, we prove that every weak Jordan *-derivation of ℜ is X -inner unless dim C ℜ C ≤ 9. Moreover, if * is of the first kind, then every weak Jordan *-derivation of ℜ is X -inner if and only if is Z (ℜ) -linear. [ABSTRACT FROM AUTHOR]