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Letℛ{\mathscr{R}}be a prime ring, which is not commutative, with involution*{*}and symmetric ring of quotientss{\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 ringℛ{\mathscr{R}}with involution*{*}admits a couple of generalized Jordan derivationsℱ1{\mathcal{F}_{1}}andℱ2{\mathcal{F}_{2}}associated with Jordan*{*}-derivations1{\mathscr{D}_{1}}and2{\mathscr{D}_{2}}such thatℱ1⁢(x)⁢x*-x⁢ℱ2⁢(x)=0{\mathcal{F}_{1}(x)x^{*}-x\mathcal{F}_{2}(x)=0}for allx∈ℛ{x\in\mathscr{R}}, then the following holds: (i) if1⁢(x)=2⁢(x){\mathscr{D}_{1}(x)=\mathscr{D}_{2}(x)}, thenℱ1⁢(x)=ℱ2⁢(x)=0{\mathcal{F}_{1}(x)=\mathcal{F}_{2}(x)=0}for allx∈ℛ{x\in\mathscr{R}}, (ii) if1⁢(x)≠2⁢(x){\mathscr{D}_{1}(x)\neq\mathscr{D}_{2}(x)}, then there existsq∈s{q\in\mathscr{Q}_{s}}such thatℱ1⁢(x)=x⁢q{\mathcal{F}_{1}(x)=xq}, andℱ2⁢(x)=q⁢x*{\mathcal{F}_{2}(x)=qx^{*}}for allx∈ℛ{x\in\mathscr{R}}. Moreover, some related results are also discussed.

georgian mathematical journal

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