Structure of generalized Jordan * * -derivations on prime * * -rings

Abstract

Abstract Let ℛ {\mathscr{R}} be a prime ring, which is not commutative, with involution * {*} and symmetric ring of quotients 𝒬 s {\mathscr{Q}_{s}} . The aim of the present paper is to describe the structures of a pair of generalized Jordan * {*} -derivations of prime * {*} -rings. Notably, we prove that if a noncommutative prime ring ℛ {\mathscr{R}} with involution * {*} admits a couple of generalized Jordan derivations ℱ 1 {\mathcal{F}_{1}} and ℱ 2 {\mathcal{F}_{2}} associated with Jordan * {*} -derivations 𝒟 1 {\mathscr{D}_{1}} and 𝒟 2 {\mathscr{D}_{2}} such that ℱ 1 ⁢ ( x ) ⁢ x * - x ⁢ ℱ 2 ⁢ ( x ) = 0 {\mathcal{F}_{1}(x)x^{*}-x\mathcal{F}_{2}(x)=0} for all x ∈ ℛ {x\in\mathscr{R}} , then the following holds: (i) if 𝒟 1 ⁢ ( x ) = 𝒟 2 ⁢ ( x ) {\mathscr{D}_{1}(x)=\mathscr{D}_{2}(x)} , then ℱ 1 ⁢ ( x ) = ℱ 2 ⁢ ( x ) = 0 {\mathcal{F}_{1}(x)=\mathcal{F}_{2}(x)=0} for all x ∈ ℛ {x\in\mathscr{R}} , (ii) if 𝒟 1 ⁢ ( x ) ≠ 𝒟 2 ⁢ ( x ) {\mathscr{D}_{1}(x)\neq\mathscr{D}_{2}(x)} , then there exists q ∈ 𝒬 s {q\in\mathscr{Q}_{s}} such that ℱ 1 ⁢ ( x ) = x ⁢ q {\mathcal{F}_{1}(x)=xq} , and ℱ 2 ⁢ ( x ) = q ⁢ x * {\mathcal{F}_{2}(x)=qx^{*}} for all x ∈ ℛ {x\in\mathscr{R}} . Moreover, some related results are also discussed.