On chains of prime ideals in rings of semialgebraic functions
Given a semialgebraic set \(M\subseteq {\mathbb R}^m\) consider the rings \(S(M)\) and \(S^*(M)\) of semialgebraic, respectively bounded semialgebraic functions on \(M\). Both of these rings are real closed in the sense of \textit{N. Schwartz} [in: Algebra and order, Proc. 1st Int. Symp., Luminy-Marseilles/France 1984, Res. Expo. Math. 14, 175--194 (1986; Zbl 0616.14019)]. This article studies (chains of) prime ideals in these rings. A main goal is to understand the structure of non-refinable chains of prime ideals of \(S^*(M)\). Some of the results generalize similar results in the the o-minimal exponentially bounded and polynomially bounded context [\textit{M. Tressl}, ``The real spectrum of continuous definable functions in o-minimal structures'', Séminaire de Structures Algébriques Ordonneés 1997-1998, 68, 1--15 (1999)].NEWLINENEWLINEHere is a selection of the main results. The height of each maximal ideal \(m\) in \(S^*(M)\) is computed by using suitable semialgebraic compactifications of \(M\). The map \(q \mapsto q \cap S^*(M)\) establishes a bijection between the sets of minimal prime ideals of \(S(M)\) and \(S^*(M)\). To investigate minimal prime ideals in these rings the author employs a finite decomposition of of any semialgebraic set \(M\) into closed pure dimensional semialgebraic subsets of different dimensions called bricks. Finally, maximal ideals of both rings \(S(M)\) and \(S^*(M)\) are studied. It is shown that for each non-compact pure dimensional semialgebraic set \(M\) of dimension \(d\) and for each \(0\leq r< d\), there exists a maximal ideal \(m\) of \(S(M)\) such that ht\((m) = r\) but ht\((m^*) = d\), where \(m^*\) is the unique maximal ideal of \(S^*(M)\) that contains \(m \cap S^*(M)\).
scientific article; zbMATH DE number 6353669