On prime ideals in rings of semialgebraic functions

Gamboa, J. M.

doi:10.1090/S0002-9939-1993-1140669-6

On prime ideals in rings of semialgebraic functions
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Proc. Amer. Math. Soc. 118 (1993), 1037-1041 Request permission

Abstract:

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.

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