On prime ideals in rings of semialgebraic functions

It is proved that if $\mathfrak{p}$ is a prime ideal in the ring $S(M)$ of semialgebraic functions on a semialgebraic set $M$, the quotient field of $S(M)/\mathfrak{p}$ 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.