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On the functionally countable subalgebra of \(C(X)\). (English) Zbl 1279.54015
For a completely regular Hausdorff space \(X\), let \(C(X)\) denote the ring of all real-valued continuous functions on \(X\). This paper deals with the subring \(C_c(X)\) of \(C(X)\) which contains all functions \(f\) in \(C(X)\) for which \(f(X)\) is countable. The authors observe that \(C_c(X)\) enjoys many properties analogous to \(C(X)\). An ideal \(I\) of \(C_c(X)\) is called a \(z_c\)-ideal if \(Z(f) \in Z_c[I]\), \(f\in C_c(X)\) \(\Rightarrow\) \(f\in I\) where \(Z_c[I] = \{Z(f) : f\in I\}\). \(z_c\)-ideals play a similar role as \(z\)-ideals in \(C(X)\).
The authors prove that (i) Every prime ideal in \(C_c(X)\) is contained in a unique maximal ideal in \(C_c(X)\). (ii) Every minimal prime ideal in \(C_c(X)\) is a \(z_c\)-ideal. (iii) Every prime ideal in \(C_c(X)\) is absolutely convex. (iv) If \(\{P_i\}_{i\in I}\) is a collection of semiprime ideals in \(C_c(X)\) such that \(P_i\) is a prime ideal for some \(i\in I\) then \(\sum_{i\in I} P_i\) is a prime ideal in \(C_c(X)\) or all of \(C_c(X)\). (v) If \(I\) is an ideal in \(C_c(X)\) then \(I\) and \(\surd I\) have the same largest \(z_c\)-ideal.
The authors show that for any space \(X\) (not necessarily completely regular), there is a zero-dimensional space \(Y\) which is a continuous image of \(X\) and \(C_c(X)\approx C_c(Y)\). It follows that for a space \(X\) with countably many components, there is a zero-dimensional space \(Y\) such that \(C_c(X)\approx C(Y)\). The authors call \(X\) a countably \(P\)-space (briefly, \(CP\)-space) if \(C_c(X)\) is a regular ring. It is shown that every \(P\)-space is a \(CP\)-space and every zero-dimensional \(CP\)-space is a \(P\)-space. Some topological and algebraic characterizations of \(CP\)-spaces are given which are parallel to the characterizations of \(P\)-spaces [L. Gillman and M. Jerison, Rings of continuous functions. Graduate Texts in Mathematics. 43. Springer-Verlag. (1976; Zbl 0327.46040)].
A commutative ring \(R\) is called self-injective (\(\aleph_0\)-selfinjective) if every homomorphism from an ideal (respectively countably generated ideal) of \(R\) into \(R\) can be extended to a homomorphism from \(R\) into \(R\). The authors prove that for a space \(X\), \(C_c(X)\) is a regular ring if and only if \(C_c(X)\) is an \(\aleph_0\)-selfinjective ring. Finally, an example of a space \(X\) is given for which \(C_c(X)\) is not isomorphic to any \(C(Y)\).

54C40 Algebraic properties of function spaces in general topology
54C30 Real-valued functions in general topology
54C05 Continuous maps
54G10 \(P\)-spaces
Full Text: DOI
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