On<i><b>C</b></i>(<i><b>X</b></i>) Modulo its Socle

Abstract

Abstract Let C F (X) denote the socle of C(X). It is shown that X is a P-space if and only if C(X) is a ℵ0-selfinjective ring or equivalently, if and only if is ℵ0-selfinjective. We also prove that X is an extremally disconnected P-space with only a finite number of isolated points if and only if is selfinjective. Consequently, if X is a P-space, then X is either an extremally disconnected space with at most a countable number of isolated points or both C(X) and have uncountable Goldie-dimensions. Prime ideals of are also studied. Keywords: Socle P-spaceSelf-injective ringExtremely disconnected spaceBasically disconnected space1991 Mathematics Subject Classification: Primary 54C40Secondary13A30 Acknowledgment The second author is partially supported by the Institute of Theoretical Physics and Mathematics (IPM).