RT Journal Article
T1 On the Krull dimension of rings of continuous semialgebraic functions
A1 Fernando Galván, José Francisco
A1 Gamboa Mutuberria, José Manuel
AB Let 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 R
PB Universidad Autónoma Madrid
SN 0213-2230
YR 2015
FD 2015
LK https://hdl.handle.net/20.500.14352/34973
UL https://hdl.handle.net/20.500.14352/34973
LA eng
NO GAAR
DS Docta Complutense
RD 9 abr 2025