# On the Krull dimension of rings of continuous semialgebraic functions

### José F. Fernando

Universidad Complutense de Madrid, Spain### José Manuel Gamboa

Universidad Complutense de Madrid, Spain

## Abstract

Let $R$ be a real closed field, $S(M)$ the ring of continuous semialgebraic functions on a semialgebraic set $M⊂R_{m}$ 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 *semi algebraic 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 $p∈M$ 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 $R$.

## Cite this article

José F. Fernando, José Manuel Gamboa, On the Krull dimension of rings of continuous semialgebraic functions. Rev. Mat. Iberoam. 31 (2015), no. 3, pp. 753–766

DOI 10.4171/RMI/852