Centralizers and Jordan triple derivations of semiprime rings

Abstract

Let R be a semiprime ring with extended centroid C and with maximal left ring of quotients $Q_{ml}(R)$. An additive map $D:R\to Q_{ml}(R)$ is called a Jordan triple derivation if $D(xyx)=D(x)yx+xD(y)x+xyD(x)$ for all $x,y\in R$. 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 $T:R\to Q_{ml}(R)$ satisfying $T(xyx)=xT(y)x$ for all $x,y\in R$ can be realized as a centralizer with only an exceptional case that $2R=0$ and R is commutative.

Keywords:

• (Semi)prime ring
• maximal left ring of quotients
• centralizer
• derivation
• Jordan triple derivation
• functional identity

2000 Mathematics Subject Classification:

• 16R60
• 16N60
• 16W25

Acknowledgements

The authors are grateful to the referee for carefully reading the manuscript.

Additional information

Funding

The work of the first author was supported in part by the Ministry of Science and Technology of Taiwan (MOST 105-2115-M-002-003-MY2) and the National Center for Theoretical Sciences (NCTS), Taipei Office.