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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 $$C(Y)$$.

##### MSC:
 54C40 Algebraic properties of function spaces in general topology 54C30 Real-valued functions in general topology 54C05 Continuous maps 54G10 $$P$$-spaces
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