connections between c(x) and c(y), where y is a subspace of x

in this paper, we introduce a method by which we can find a close connection between the set of prime $z$-ideals of $c(x)$ and the same of $c(y)$, for some special subset $y$ of $x$. for instance, if $y=coz(f)$ for some $fin c(x)$, then there exists a one-to-one correspondence between the set of prime $z$-ideals of $c(y)$ and the set of prime $z$-ideals of $c(x)$ not containing $f$. moreover, considering these relations, we obtain some new characterizations of classical concepts in the context of $c(x)$. for example, $x$ is an $f$-space if and only if the extension $phi : beta yrightarrowbeta x$ of the identity map $imath: yrightarrow x$ is one-to-one, for each $z$-embedded subspace $y$ of $x$. supposing $p$ is a non-isolated $g_delta$-point in $x$ and $y=xsetminus{p}$, we prove that $m^p(x)$ contains no non-trivial maximal $z$-ideal if and only if $pinbe x$ is a quasi $p$-point if and only if each point of $beta y setminus y$ is a $p$-point with respect to $y$.