Mulero, M. A. Algebraic properties of rings of continuous functions. (English) Zbl 0840.54020 Fundam. Math. 149, No. 1, 55-66 (1996). Denote by \(C(X)\) (resp. \(C^* (X)\)) the ring of real-valued continuous (resp. continuous bounded) functions defined on a topological space. The author proves that if a continuous map \(X\to S\) is open and closed (resp. open and proper) then going-up and going-down theorems hold for \(C^* (S)\to C^* (X)\) (resp. for \(C(S)\to C(X)\)). Under the same hypothesis he proves that the prime spectrum \(\text{Spec} (C^* (X))\to \text{Spec} (C^*(X))\) is open and closed. In another part of the paper he shows that if every closed set in \(X\) is a zero-set then a \(C(X)\)- module of finite presentation is flat if and only if it is torsion-free. Reviewer: Y.Felix (Louvain-La-Neuve) Cited in 1 ReviewCited in 18 Documents MSC: 54C40 Algebraic properties of function spaces in general topology 13B24 Going up; going down; going between (MSC2000) 13C11 Injective and flat modules and ideals in commutative rings Keywords:primary ideal; flat module PDFBibTeX XMLCite \textit{M. A. Mulero}, Fundam. Math. 149, No. 1, 55--66 (1996; Zbl 0840.54020) Full Text: EuDML