I am reading the proof of the Real Nullstellensatz (from the Real Algebraic Geometry book, by Bochnak, Coste and Roy). I add a picture of this proof
And there is a notation I had not seen before: the ring of fractions $B_{\overline{P}}$. I would appreciate any help to understand this. Thank you!
To clear this from the unanswered queue, this is the localization of $B$ at the element $\overline{P}$.
The localization of a (commutative) ring at a multiplicatively closed subset $S$ is the set of pairs $(r,s)$ with addition defined by $(r,s)+(r',s')=(rs'+r's,ss')$, multiplication defined by $(r,s)\cdot(r',s')=(rr',ss')$, and two pairs $(r,s)$ and $(r',s')$ are equal iff there exists some $t\in S$ so that $t(rs'-r's)=0$.
This specific example is the case when $S=\{1,\overline{P},\overline{P}^2,\overline{P}^3,\cdots\}$ and $R=B$. In the language of geometry, this corresponds to considering the open subset/subvariety/subscheme which is the complement of the vanishing locus of $\overline{P}$.