3) Geometric interpretation of the localization. Let V be an irreducible algebraic variety. Then P = J(V) is a prime ideal of C [ X 1 , . . , X n ] {\displaystyle \mathbb {C} [X_{1},...,X_{n}]} and so C [ V ] = C [ X 1 , . . , X n ] / J ( V ) {\displaystyle \mathbb {C} [V]=\mathbb {C} [X_{1},...,X_{n}]/J(V)} is an integral domain. The localization C [ X 1 , . . , X n ] P {\displaystyle \mathbb {C} [X_{1},...,X_{n}]_{P}} is a subring of C ( X 1 , . . , X n ) {\displaystyle \mathbb {C} (X_{1},...,X_{n})} consisting of rational functions { f / g : f , g ∈ C [ X 1 , . . , X n ] , g ∉ P } {\displaystyle \{f/g:f,g\in \mathbb {C} [X_{1},...,X_{n}],g\notin P\}} which are defined on a nonempty subset of V. If V = {x} is a point, then P is maximal and C [ X 1 , . . , X n ] P = { f / g : f , g ∈ C [ X 1 , . . , X n ] , g ( x ) ≠ 0 } {\displaystyle \mathbb {C} [X_{1},...,X_{n}]_{P}=\{f/g:f,g\in \mathbb {C} [X_{1},...,X_{n}],g(x)\neq 0\}} consists of rational functions which are defined at x.
2007, Ivan Fesenko, “Rings and modules”, in G13ALS Algebra 2, 2007/2008 @ maths.nottingham.ac.uk, page 27