On chains of prime ideals in rings of semialgebraic functions

Abstract

In this work, we study the structure of non-refinable chains of prime ideals in the (real closed) rings s(M) and s*(M) of semialgebraic and bounded semialgebraic functions on a semialgebraic set M⊂ℝm. We pay special attention to the prime z-ideals of s(M) and the minimal prime ideals of both rings. For the last, a decomposition of each semialgebraic set as an irredundant finite union of closed pure dimensional semialgebraic subsets plays a crucial role. We prove moreover the existence of maximal ideals in the ring s(M) of prefixed height whenever M is non-compact.