On the Krull dimension of rings of continuous semialgebraic functions

Resumen

Let R be a real closed field, S(M) the ring of continuous semialgebraic functions on a semialgebraic set M⊂Rm 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 semi algebraic depth of a prime ideal pp 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∈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 R.