Weak Jordan *-derivations of prime rings

Abstract

Let * be an involution of a non-commutative prime ring [Formula: see text] with the maximal symmetric ring of quotients and the extended centroid of [Formula: see text] denoted by [Formula: see text] and [Formula: see text], respectively. Consider [Formula: see text] be an additive map, if [Formula: see text] for all [Formula: see text], then such a map [Formula: see text] 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 [Formula: see text] is [Formula: see text]-inner unless [Formula: see text]. Moreover, if * is of the first kind, then every weak Jordan *-derivation [Formula: see text] of [Formula: see text] is [Formula: see text]-inner if and only if [Formula: see text] is [Formula: see text]-linear.