Certain basic functional identities of semiprime rings

Abstract

The goal of the article is to discuss certain basic functional identities of semiprime rings. Specifically, the following result is proved: Let R be a semiprime ring with extended centroid C and let $f,g:R\to Q_{ml}(R)$ be additive maps, where $Q_{ml}(R)$ denotes the maximal left ring of quotients of R. Suppose that $f(x)x+xg(x)\in C$ for all $x\in R$. Then there exist $b\in Q_{ml}(R),\beta \in C$ and additive maps $\nu ,\eta :R\to C$ such that $f(x)=xb+\nu (x)$ and $g(x)=(-b+\beta )x+\eta (x)$ for all $x\in R$. Moreover, some related functional identities are also characterized.

keywords:

• (Semi)prime ring
• Posner’s second theorem
• extended centroid
• orthogonal completion
• functional identity

2000 Mathematics Subject Classification:

• 16R60
• 16N60
• 16W10
• 16W25

Acknowledgements

The author is grateful to the referee for carefully reading the manuscript and also thanks Professor M. Brešar for putting forward some constructive opinions.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

The work 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.