Pseudo-valuation rings and \(C(X)\)
A pseudo-valuation domain (in short a PVD) is a local integral domain \(D\) with maximal ideal \(M\) and field of fractions \(K\) such that the set \(M:M=\{x\in K\; ; \; xM\subseteq M\}\) is a valuation domain. The ring \(M:M\) being a valuation domain is equivalent to the following condition: for every \(a,b\) in \(D\) and \(c\) in \(M\), \(a\) divides \(bc\) or \(b\) divides \(ac\) (note that \(D\) is a valuation domain if, and only if, for every \(a,b\) in \(D\), \(a\) divides \(b\) or \(b\) divides \(a\)). After a survey where they are placed in a more general context, the authors consider the ring \(C(\Omega)\) of real valued continuous functions on a compact Hausdorff space \(\Omega\). Let \(P\) be a prime ideal of \(C(\Omega)\). The aim of this paper is to investigate when \(C(\Omega)/P\) is a valuation domain or a pseudo-valuation domain. Now, \(C(\Omega)/P\) is a bounded real closed domain. So the author study more generally the case of bounded real closed domain. Before to state the main results of this paper, we need to list some definitions. An \(f\)-ring is a lattice-ordered ring which is a subdirect product of totally ordered rings. The set of bounded elements of an \(f\)-ring \(A\), with identity element \(1\), is the set \(A^*=\{a\in A\; ; \; \exists n\in \mathbb{N},\; |a|\leq n\}\). If \(A=A^*\), then \(A\) is said to be bounded. Let \(m\in \mathbb{N}\); an \(f\)-ring \(A\) is \(m\) convex if, for every \(a,b\) in \(A\) such that \(0\leq a\leq b^m\), \(b\) divides \(a\). Now, a real closed domain is a domain \(D\) which satisfies the following conditions. For each prime ideal \(Q\), the domain \(D/Q\) is integrally closed, and its quotient field is a real closed field; the squares of \(D\) form a positive cone of a partial order which makes \(D\) into a \(2\)-convex \(f\)-ring. A previous result showed that, for every \(a,b\) in a real closed domain, there exists \(n\in \mathbb{N}\) such that \(a\) divides \(b\) or \(b\) divides \(a^n\). If \(K\) is the quotient field of a bounded real closed domain \(D\), then the set \(K^*\) of bounded elements of \(K\) is called the set of finite elements, it is a valuation domain, with maximal ideal \(K^0=\{a\in K\; ; \; \forall n\in \mathbb{N} \; n|a|\leq 1\}\), which is called the set of infinitely small elements. The first result of this paper states that if \(D\) is a bounded real closed domain with maximal ideal \(M\) and quotient field \(K\), then \(D\) is a valuation domain if, and only if, one of the following three equivalent conditions hold. \(K^*=D\); \(K^0=M\) and the natural map \(D/M\rightarrow K^*/K^0\) is surjective; \(D\) is \(1\)-convex. The second result asserts that, with the same hypothesis, \(D\) is a PVD if, and only if, one of the following five equivalent conditions hold. \(M\) is an interval of \(K\); \(M\) is an interval of \(K^*\); \(M\) is an ideal of \(K^*\); \(K^*\subseteq M:M\); \(M:M\) is a valuation domain. The authors deduce some corollaries and they provide examples. Finally, they give an example which proves that if \(\Omega\) is a compact hausdorff space and \(P\) is a prime ideal of \(C(\Omega)\), then \(C(\Omega)/P\) need not be a PVD, and an example where \(C(\Omega)/P\) is a PVD but not a valuation domain.
scientific article; zbMATH DE number 6915740
scientific article