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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] and so ℂ[V]=ℂ[X_1,...,X_n]/J(V) is an integral domain. The localization ℂ[X_1,...,X_n]_P is a subring of ℂ(X_1,...,X_n) consisting of rational functions f/g:f,g∈ℂ[X_1,...,X_n],g∉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) ne 0 consists of rational functions which are defined at x.

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