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Algebraic properties of rings of continuous functions. (English) Zbl 0840.54020

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.

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
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