More on the Generalized (m,n)-Jordan Derivations and Centralizers on Certain Semiprime Rings
Abstract: In this paper we give an affirmative answer to two conjectures on generalized (m, n)-Jordan derivations and generalized (m, n)-Jordan centralizers raised in [S. Ali and A. Fošner, On Generalized (m, n)-Derivations and Generalized (m, n)-Jordan Derivations in Rings, Algebra Colloq. 21 (2014), 411-420] and [A. Fošner, A note on generalized (m, n)-Jordan centralizers, Demonstratio Math. 46 (2013), 254-262]. Precisely, when R is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonze… Show more
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On the characterization of generalized (m, n)-Jordan *-derivations in prime rings
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.
On the characterization of generalized (m, n)-Jordan *-derivations in prime rings
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.
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