: A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. The main purpose of this paper is to prove a *-version of Posner's theorem mentioned above. Moreover, we describe the structure of an arbitrary additive mapping which is *-centralizing on a prime ring with involution.

In [Acta Math. Hungar. 66 (1995), 337–343], Bell and Daif proved that if R is a prime ring admitting a nonzero derivation such that d ( x y ) = d ( y x ) ${d(xy)=d(yx)}$ for all x , y ∈ R ${x,y\in R}$ , then R is commutative. The objective of this paper is to examine similar problems when the ring R is equipped with involution. It is shown that if a prime ring R with involution * of a characteristic different from 2 admits a nonzero derivation d such that d ( x x * ) = d ( x * x ) ${d(xx^*)=d(x^*x)}$ for all x ∈ R and S ( R ) ∩ Z ( R ) ≠ ( 0 ) ${S(R)\cap Z(R)\ne (0)}$ , then R is commutative. Moreover, some related results have also been discussed.

Let R be a ring and S a nonempty subset of R. Suppose that θ and ϕ are endomorphisms of R. An additive mapping δ : R → R is called a left (θ, ϕ)‐derivation (resp., Jordan left (θ, ϕ)‐derivation) on S if δ(xy) = θ(x)δ(y) + ϕ(y)δ(x) (resp., δ(x²) = θ(x)δ(x) + ϕ(x)δ(x)) holds for all x, y ∈ S. Suppose that J is a Jordan ideal and a subring of a 2‐torsion‐free prime ring R. In the present paper, it is shown that if θ is an automorphism of R such that δ(x²) = 2θ(x)δ(x) holds for all x ∈ J, then either J⫅Z(R) or δ(J) = (0). Further, a study of left (θ, θ)‐derivations of a prime ring R has been made which acts either as a homomorphism or as an antihomomorphism of the ring R.