A Note on Jordan Triple Higher *-Derivations on Semiprime Rings
Abstract: We introduce the following notion. Let N 0 be the set of all nonnegative integers and let = ( ) ∈N 0 be a family of additive mappings of a * -ring R such that 0 = ; D is called a Jordan higher * -derivation (resp., a Jordan higher) for all , ∈ and each ∈ N 0 . It is shown that the notions of Jordan higher * -derivations and Jordan triple higher * -derivations on a 6-torsion free semiprime * -ring are coincident.
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Cited by 3 publications
References 11 publications
We obtain the following result: Let R be a 2-torsion free semiprime ∗-ring. Suppose there exists a family of additive mappings F=(fi)i∈ℕ0 of ℝ associated with some higher derivation D = (di)i∈ℕ0 of R, such that f0 = idR and the relation fn(xx∗) = ∑i+j=nfi(x)dj(x∗) is fulfilled for each n ∈ ℕ0 and for all x ∈ R. Then F is a generalized higher derivation. Further, we also prove that if the relation fn(xy∗x) = ∑i+j+k=n fi(x)dj(y∗)dk(x) is fulfilled for each n ∈ ℕ0 and for all x,y ∈ R, then F is also a generalized higher derivation.
We obtain the following result: Let R be a 2-torsion free semiprime ∗-ring. Suppose there exists a family of additive mappings F=(fi)i∈ℕ0 of ℝ associated with some higher derivation D = (di)i∈ℕ0 of R, such that f0 = idR and the relation fn(xx∗) = ∑i+j=nfi(x)dj(x∗) is fulfilled for each n ∈ ℕ0 and for all x ∈ R. Then F is a generalized higher derivation. Further, we also prove that if the relation fn(xy∗x) = ∑i+j+k=n fi(x)dj(y∗)dk(x) is fulfilled for each n ∈ ℕ0 and for all x,y ∈ R, then F is also a generalized higher derivation.
In this article, we define the following: Let N 0 {{\mathbb{N}}}_{0} be the set of all nonnegative integers and D = ( d i ) i ∈ N 0 D={\left({d}_{i})}_{i\in {{\mathbb{N}}}_{0}} a family of additive mappings of a ∗ \ast -ring R R such that d 0 = i d R {d}_{0}=i{d}_{R} . D D is called a Jordan ( α , β ) \left(\alpha ,\beta ) -higher ∗ \ast -derivation (resp. a Jordan triple ( α , β ) \left(\alpha ,\beta ) -higher ∗ \ast -derivation) of R R if d n ( a 2 ) = ∑ i + j = n d i ( β j ( a ) ) d j ( α i ( a ∗ i ) ) {d}_{n}\left({a}^{2})={\sum }_{i+j=n}{d}_{i}\left({\beta }^{j}\left(a)){d}_{j}\left({\alpha }^{i}\left({a}^{{\ast }^{i}})) (resp. d n ( a b a ) = ∑ i + j + k = n d i ( β j + k ( a ) ) d j ( β k ( α i ( b ∗ i ) ) ) d k ( α i + j ( a ∗ i + j ) ) {d}_{n}\left(aba)={\sum }_{i+j+k=n}{d}_{i}\left({\beta }^{j+k}\left(a)){d}_{j}\left({\beta }^{k}\left({\alpha }^{i}\left({b}^{{\ast }^{i}}))){d}_{k}\left({\alpha }^{i+j}\left({a}^{{\ast }^{i+j}})) ) for all a , b ∈ R a,b\in R and each n ∈ N 0 n\in {{\mathbb{N}}}_{0} . We show that the two notions of Jordan ( α , β ) \left(\alpha ,\beta ) -higher ∗ \ast -derivation and Jordan triple ( α , β ) \left(\alpha ,\beta ) -higher ∗ \ast -derivation on a 6-torsion free semiprime ∗ \ast -ring are equivalent.
We investigate the additivity and multiplicativity of centrally extended higher derivations and show that every centrally extended higher derivation of a semiprime ring with no nonzero central ideals is a higher derivation. Moreover, we study preservation of the center of the ring by a centrally extended higher derivation.
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