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

Abstract

. In this article we consider some relations between the topological properties of the spaces 𝑋 and 𝑀𝑖𝑛 ( 𝐶 𝑐 ( 𝑋 )) with algebraic properties of 𝐶 𝑐 ( 𝑋 ) . We observe that the compactness of 𝑀𝑖𝑛 ( 𝐶 𝑐 ( 𝑋 )) is equivalent to the von-Neumann regularity of 𝑞 𝑐 ( 𝑋 ) , the classical ring of quotients of 𝐶 𝑐 ( 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 𝐶 𝑐 ( 𝑋 ) and in this case 𝑀𝑖𝑛 ( 𝐶 ( 𝑋 )) and 𝑀𝑖𝑛 ( 𝐶 𝑐 ( 𝑋 )) are homeomorphic spaces. We also observe that if 𝑋 is an 𝐹 𝑐 -space, then 𝑀𝑖𝑛 ( 𝐶 𝑐 ( 𝑋 )) is compact if and only if 𝑋 is countably basically disconnected if and only if 𝑀𝑖𝑛 ( 𝐶 𝑐 ( 𝑋 )) is homeomorphic with 𝛽 0 𝑋 . Finally, by introducing 𝑧 ◦ 𝑐 -ideals, countably cozero complemented spaces, we obtain some conditions on 𝑋 for which 𝑀𝑖𝑛 ( 𝐶 𝑐 ( 𝑋 )) becomes compact, basically disconnected and extremally disconnected.