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

Abstract

Let

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.