On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋)

In this article we consider some relations between the topological properties of the spaces X and  Min(Cc (X)) with algebraic properties of Cc (X). We observe that the compactness of  Min(Cc (X)) is equivalent to the von-Neumann regularity of  qc (X), the classical ring of quotients of Cc (X). Furthermore, we show that if 𝑋 is a strongly zero-dimensional space, then each contraction of a minimal prime ideal of 𝐶(𝑋) is a minimal prime ideal of Cc(X) and in this case 𝑀𝑖𝑛(𝐶(𝑋)) and Min(Cc (X)) are homeomorphic spaces. We also observe that if 𝑋 is an Fc-space, then  Min(Cc (X)) is compact if and only if 𝑋 is countably basically disconnected if and only if Min(Cc(X)) is homeomorphic with β۰X. Finally, by introducing zoc-ideals, countably cozero complemented spaces, we obtain some conditions on X for which  Min(Cc (X)) becomes compact, basically disconnected and extremally disconnected.

Keywords:

The space of minimal prime ideals , strongly zero-dimensional space , countably basically disconnected space , countably cozero complemented space , z^۰_c-ideals