### 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*).

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