R-P-Spaces and Subrings of C(X)

Abstract

A Tychono
space X is called a P-space, if M^p = O^p for each

p ∈ X. For each subring R of C(X), we call X an R-P-space, if

M^p ∩ R = O^p ∩ R for each p ∈ X. By constructing the smallest

invertible subring of C(X) in which a given subring R is embedded,

various characterizations of R-P-space are investigated. Moreover, using some properties of the mappings Z_A and ℑ_A, we provide shorter

proofs to some main results of W. Murrqy, J. Sack and S. Watson, P- spaces and intermediate rings of continuous functions, Rocky Mount.

Math., to appers. Also, by investigating a characterization of C(X)

among its intermediate subrings, we approach to some other results of

the mentioned paper by a di
erent way. Finally, by giving a charac-

terzation of intermediate C-rings of C(X) (subring of C(X) containing

C(X) which are isomorphic with C(Y ) for some Tychono
space Y ),

some characterizations of C(X) among its intermediate C-rings, when-

ever X is a P-space, are established. As an special case, we consider

intermediate C-rings of the form I + C(X) where I is an ideal in

C(X).