$P$-spaces and intermediate rings of continuous functions

R-P-spaces and subrings of C(X)

“…By the following statement, we give some algebraic characterizations of P-spaces via intermediate rings of C(X). This statement is also a generalization of [14,Theorem 2.5]. Recall that we use M…”

“…(e⇒a) This is evident, since, S C ( f ) = cl βX Z( f ) for each f ∈ C(X). In [14,Proposition 3.6], it is shown that whenever A(X) is an intermediate ring of C(X) such that A(X) C(X), then there exists a non-maximal prime ideal in A(X). Using Theorem 3.3, we give a different proof to this statement.…”

“…Some algebraic characterizations of P-spaces via intermediate rings follows from this fact. Moreover, by establishing a characterization of C(X) among its intermediate rings, some results of [14] are proved by a different way. In Section 4, we study R-P-spaces with the restriction that R is an intermediate C-ring of C(X).…”

“…[14, Proposition 3.6] Let A(X) be an intermediate ring of C(X) such that A(X) C(X). Then there exists a non-maximal prime ideal in A(X).Proof.…”

On the mappings ${\mathcal Z}_A$ and $\Im_A$ in intermediate rings of $C(X)$

“…A is not a Z A -ideal in A(X), however, it is clearly a z-ideal. In [7,Theorem 2.14] it is stated that whenever every ideal of an intermediate ring A(X) is a Z A -ideal, then X is a P -space. The next theorem shows that even when every z-ideal is a Z A -ideal, then we have A(X) = C(X).…”