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 vonNeumann regularity of  qc (X), the classical ring of quotients of Cc (X). Furthermore, we show that if 𝑋 is a strongly zerodimensional 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 Fcspace, 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 β0X. Finally, by introducing 𝑧◦𝑐-ideals, countably cozero complemented spaces, we obtain some conditions on X for which  Min(Cc (X)) becomes compact, basically disconnected and extremally disconnected.