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 $\ast$-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 $\ast$-derivation) $\delta:R\to R$ such that \[ [\delta(x),x]\in Z(R)~~(resp.~~[\delta(x),x^{\ast}]\in Z(R))\text{~for~all~}x\in R, \] where $'\ast'$ is an involution on $R,$ then $R$ is an order in a central simple algebra of dimension at most 4 over its center.

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Let R be a prime ring and S a non-empty subset of R. Suppose that θ, φ are endomorphisms of R. An additive mappingSuppose that U is a Lie ideal of R such that u 2 ∈ U , for all u ∈ U . The main result of the present paper states that if F is a generalized (θ, θ)-derivation on U which also acts as a homomorphism or as an anti-homomorphism on U , then either d = 0 or U ⊆ Z(R). Mathematics Subject Classification: 16W25,16N60, 16U80

Let R be a ring. A map F : R → R (not necessarily additive) is called a multiplicative (generalized