Characterizing Jordan n -Derivations of Unital Rings Containing Idempotents
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Characterizing Jordan n-Derivations of Unital Rings Containing Idempotents Xiaofei Qi1 · Zhiling Guo1 · Ting Zhang1 Received: 19 August 2019 / Revised: 28 November 2019 / Accepted: 28 December 2019 © Iranian Mathematical Society 2020
Abstract Assume that R is a unital ring containing a nontrivial idempotent. By introducing the concept of Jordan n-derivations (n is any positive integer), it is shown that, under certain conditions, every multiplicative (without any linearity and additivity) Jordan n-derivation δ on R is additive; and furthermore, δ is a Jordan derivation if the characteristic of R is not n − 1 and δ is a generalized Jordan derivation if the characteristic of R is n − 1 (n > 3). Based on these results, it turns out that a map on R is a multiplicative Jordan n-derivation if and only if it is an additive Jordan derivation. As applications, multiplicative Jordan n-derivations on triangular rings, prime rings, matrix algebras, nest algebras and von Neumann algebras are, respectively, completely characterized, which generalize some known related results. Keywords Jordan derivation · Derivation · Generalized derivations · Unital rings · Matrix algebras Mathematics Subject Classification 16W10 · 47B47
1 Introduction Let R be an associate algebra or ring. Denote by {x, y} = x y + yx the Jordan product of elements x, y ∈ R. A linear (or an additive) map δ : R → R is called
Communicated by Shirin Hejazian. This work is partially supported by National Natural Science Foundation of China (11671006) and Outstanding Youth Foundation of Shanxi Province (201701D211001).
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Xiaofei Qi [email protected] Zhiling Guo [email protected] Ting Zhang [email protected]
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School of Mathematical Science, Shanxi University, Taiyuan 030006, People’s Republic of China
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Bulletin of the Iranian Mathematical Society
a derivation if δ(x y) = δ(x)y + xδ(y) holds for all x, y ∈ R; is called a Jordan derivation if δ({x, y}) = {δ(x), y} + {x, δ(y)} holds for all x, y ∈ R, or equivalently, if δ(x 2 ) = δ(x)x + xδ(x) holds for all x ∈ R in the case that the characteristic of R is not 2; is called a Jordan triple derivation if δ(x yx) = δ(x)yx + xδ(y)x + x yδ(x) (or equivalently, δ(x yz + zyx) = δ(x)yz + xδ(y)z + x yδ(z)+δ(z)yx + zδ(y)x + zyδ(x)) holds for all x, y, z ∈ R. The problem of whether or not linear or additive Jordan (triple) derivations are derivations had received many mathematicians’ attention for many years (see [1– 3,5,8,10,12,14–16,21] and references therein). In Ref. [10], Herstein first proved that every additive Jordan derivation from a prime ring of characteristic not 2 into itself is a derivation. Thereafter, Brešar [2] extended Herstein’s theorem to 2-torsion free semiprime ring. Li and Lu [14] proved that every additive Jordan derivation on reflexive algebras is an additive derivation. Benkovicˇ [3] showed that every linear Jordan derivation from upper triangular matrix algebras into its bimodule is the sum of a linear derivation and a linear antiderivation. In Ref. [2], Brešar proved that
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