On the characterization of generalized (m, n)-Jordan *-derivations in prime rings

Abstract

Let 𝒜 {\mathcal{A}} be a prime ring equipped with an involution ‘ * {*} ’ of order 2 and let m ≠ n {m\neq n} be some fixed positive integers such that 𝒜 {\mathcal{A}} is 2 ⁢ m ⁢ n ⁢ ( m + n ) ⁢ | m - n | {2mn(m+n)|m-n|} -torsion free. Let 𝒬 m ⁢ s ⁢ ( 𝒜 ) {\mathcal{Q}_{ms}(\mathcal{A})} be the maximal symmetric ring of quotients of 𝒜 {\mathcal{A}} and consider the mappings ℱ {\mathcal{F}} and 𝒢 : 𝒜 → 𝒬 m ⁢ s ⁢ ( 𝒜 ) {\mathcal{G}:\mathcal{A}\to\mathcal{Q}_{ms}(\mathcal{A})} satisfying the relations ( m + n ) ⁢ ℱ ⁢ ( a 2 ) = 2 ⁢ m ⁢ ℱ ⁢ ( a ) ⁢ a * + 2 ⁢ n ⁢ a ⁢ ℱ ⁢ ( a ) (m+n)\mathcal{F}(a^{2})=2m\mathcal{F}(a)a^{*}+2na\mathcal{F}(a) and ( m + n ) ⁢ 𝒢 ⁢ ( a 2 ) = 2 ⁢ m ⁢ 𝒢 ⁢ ( a ) ⁢ a * + 2 ⁢ n ⁢ a ⁢ ℱ ⁢ ( a ) (m+n)\mathcal{G}(a^{2})=2m\mathcal{G}(a)a^{*}+2na\mathcal{F}(a) for all a ∈ 𝒜 {a\in\mathcal{A}} . Using the theory of functional identities and the structure of involutions on matrix algebras, we prove that if ℱ {\mathcal{F}} and 𝒢 {\mathcal{G}} are additive, then 𝒢 = 0 {\mathcal{G}=0} . We also show that, in case ‘ * * ’ is any nonidentity anti-automorphism, the same conclusion holds if either ‘ * {*} ’ is not identity on 𝒵 ⁢ ( 𝒜 ) {\mathcal{Z}(\mathcal{A})} or 𝒜 {\mathcal{A}} is a PI-ring.