Gamboa Mutuberria, José Manuel2023-06-202023-06-201993J. Bochnak, M. Coste, and M. F. Roy, Geometric algebrique reelle, Ergeb. Math. Grenzgeb.(3), vol. 12, Springer-Verlag, Berlin and New York, 1987. M. Carral and M. Coste, Normal spectral spaces and their dimensions, J. Pure Appl. Algebra 301 (1983), 227-235. 3. J. M. Gamboa and J. M. Ruiz, On rings of semialgebraic functions, Math. Z. 206 (1991), 527-532. M. Henriksen and J. R. Isbell, On the continuity of the real roots of an algebraic equation, Proc. Amer. Math. Soc. 4 (1953), 431-434. J. R. Isbell, More on the continuity of the real roots of an algebraic equation, Proc. Amer. Math. Soc. 5(1954), 439. T. Recio, Una descomposicion de un conjunlo semialgebraico, Proc. A.M.E.L. Mallorca, 1977. J. J. Risler, Le theoreme des zeros en geometries algebrique et analytique relies, Bull. Soc. Math. France 104 (1976), 113-127. J. M. Ruiz, Cones locaux et completions, C.R. Acad. Sci. Paris Ser. I Math. 302 (1986), 67-69. N. Schwartz, The basic theory of real closed spaces, Mem. Amer. Math. Soc, no. 397, Amer. Math. Soc, Providence, RI, 1989.0002-993910.2307/2160055https://hdl.handle.net/20.500.14352/57279It is proved that if p is a prime ideal in the ring S{M) of semialgebraic functions on a semialgebraic set M, the quotient field of S(M)/p is real closed. We also prove that in the case where M is locally closed, the rings S(M) and P(M)—polynomial functions on M—have the same Krull dimension. The proofs do not use the theory of real spectra.engjournal articlehttp://www.ams.org/journals/proc/1993-118-04/S0002-9939-1993-1140669-6/S0002-9939-1993-1140669-6.pdfhttp://www.ams.orgrestricted access512.7Prime Ideal In The Ring Of Semialgebraic FunctionsKrull DimensionGeometria algebraica1201.01 Geometría Algebraica