In this paper, we consider skew Lie product on an involutive ring and study several algebraic identities for it, which include generalized derivations of the ring. The results give information about the commutativity of the ring and a description of the generalized derivations.

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Let R be any ring containing a non-tivial idempotent element e. Let ℑ : R → R be a mapping such that ℑ(ab) = ℑ(b)a + bℑ(a) for all a, b ∈ R. In this note, our aim is to show that under some suitable restrictions imposed on R, ℑ is additive.

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Let R be a ring and Z(R) be the center of R. The aim of this paper is to define the notions of centrally extended Jordan derivations and centrally extended Jordan * -derivations, and to prove some results involving these mappings. Precisely, we prove that if a 2-torsion free noncommutative prime ring R admits a centrally extended Jordan derivation (resp. centrally extended Jordan * -derivation)) for all x ∈ R, where ′ * ′ is an involution on R, then R is an order in a central simple algebra of dimension at most 4 over its center.