A note on a sheaf of real regular functions

Abstract

Coste and Roy in 1979 defined a structural sheaf on the real Zariski spectrum of a semi-real ring $A$ and asked whether the ring of the global sections is ${\sum }_1^{-1}A$ where ${\sum }_1$ is the multiplicative subset $\{1+{\sum }_{i=1}^n{a}_i^2|a_i\in A,n\in N\}$ of $A$. We give a positive answer to this question for real rings and integral domains.