Rings of quotients of rings of functions
Let \(X\) be a Tychonoff space and \(C(X)\) denote the ring of continuous real valued functions on \(X\). A zero-set \(Z\subset X\) is said to be \(z\)-complemented in \(X\) in case there is a zero-set \(\widehat Z\subset X\) that \(Z\cup \widehat Z=X\) and \(Z\cap \widehat Z\) is nowhere dense in \(X\). The space \(X\) is called \(z\)-good in case every zero-set is \(z\)-complemented; otherwise \(X\) is \(z\)-bad. In the present paper, the authors construct various examples of \(z\)-good and \(z\)-bad spaces. Among the \(z\)-good spaces are all metric spaces and all spaces with the property that the closure of every cozero-set in \(X\) is a zero-set. Another result says that if \(M\) is a compact metric space and every zero-set in \(X\) has non-empty interior, then \(M\times X\) is \(z\)-good if, and only if, \(X\) is a \(P\)-space. For any filter \(F\) of open subsets of \(X\) we define the ring of partial functions \(C_F(X) =\bigcup\{C(U): U\in F\}\), where \(f\equiv g\) for \(f\in C(U)\) and \(g\in C(V)\) in case \(f|W = g|W\) for some \(W\in F\) with \(W\subset U\cap V\). A classical result in the book ``Rings of quotients of rings of functions'' by \textit{N. J. Fine, L. Gillman} and \textit{J. Lambek} [(McGill Univ. Press, Montreal) (1965; Zbl 0143.35704)] says that \(C_F(X)\) is isomorphic to the total ring of quotients on \(C(X)\) in case \(F\) is the filter of dense open subsets of \(X\). The present paper generalizes this result: first, to the case where \(F\) is an arbitrary filter of open subsets and second, to the case where \(F\) is a Gabriel filter. Some open questions are also raised.
scientific article; zbMATH DE number 2136469