Centralizers and Jordan triple derivations of semiprime rings. Issue 1 (2nd January 2019)

• Record Type: Journal Article
• Authors: Lee, Tsiu-Kwen; Quynh, Truong Cong
• Is Part Of: Communications in algebra. Volume 47:Issue 1(2019)
• Page Start: 236
• Page End: 251
• Publication Date: 2019-01-02
• Subjects: (Semi)prime ring -- maximal left ring of quotients -- centralizer -- derivation -- Jordan triple derivation -- functional identity; 16R60 -- 16N60 -- 16W25; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2018.1472275
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: Let R be a semiprime ring with extended centroid C and with maximal left ring of quotients Q m l ( R ) . An additive map D : R → Q m l ( R ) is called a Jordan triple derivation if D ( x y x ) = D ( x ) y x + x D ( y ) x + x y D ( x ) for all x, y ∈ R . In 1957, Herstein proved that a Jordan triple derivation, which is also a Jordan derivation, of a noncommutative prime ring of characteristic 2, must be a derivation. In 1989, Brešar proved that any Jordan triple derivation of a 2-torsion free semiprime ring is a derivation. In the article, we give a complete characterization of Jordan triple derivations of arbitrary semiprime rings. To get such a characterization we first show that, in some sense, an additive map T : R → Q m l ( R ) satisfying T ( x y x ) = x T ( y ) x for all x, y ∈ R can be realized as a centralizer with only an exceptional case that 2 R = 0 and R is commutative.