Fernando Galván, José FranciscoGamboa Mutuberria, José Manuel2023-06-192023-06-1920150213-223010.4171/RMI/852https://hdl.handle.net/20.500.14352/34973Let R be a real closed field, S(M) the ring of continuous semialgebraic functions on a semialgebraic set M subset of R-m and S* (M) its subring of continuous semialgebraic functions that are bounded with respect to R. In this work we introduce semialgebraic pseudo-compactifications of M and the semialgebraic depth of a prime ideal p of S(M) in order to provide an elementary proof of the finiteness of the Krull dimensions of the rings S(M) and S* (M) for an arbitrary semialgebraic set M. We are inspired by the classical way to compute the dimension of the ring of polynomial functions on a complex algebraic set without involving the sophisticated machinery of real spectra. We show dim(S(M)) = dim(S* (M)) = dim(M) and prove that in both cases the height of a maximal ideal corresponding to a point p is an element of M coincides with the local dimension of M at p. In case p is a prime z-ideal of S(M), its semialgebraic depth coincides with the transcendence degree of the real closed field qf(S(M)/p) over Rengjournal articlehttp://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=31&iss=3&rank=1http://arxiv.org/abs/1306.4109v1http://www.ems-ph.org/open access512.7Semialgebraic functionbounded semialgebraic functionz-idealsemialgebraic depthKrull dimensionlocal dimensiontranscendence degreereal closed ringreal closed fieldreal closure of a ringGeometria algebraica1201.01 Geometría Algebraica