Some new results on functions in $C(X)$ having their support on ideals of closed sets

For any ideal $mathcal{P}$ of closed sets in $X$, let $C_mathcal{P}(X)$ be the family of those functions in $C(X)$ whose support lie on $mathcal{P}$. Further let $C^mathcal{P}_infty(X)$ contain precisely those functions $f$ in $C(X)$ for which for each $epsilon >0, {xin X: lvert f(x)rvertgeq epsilon}$ is a member of $mathcal{P}$. Let $upsilon_{C_{mathcal{P}}}X$ stand for the set of all those points $p$ in $beta X$ at which the stone extension $f^*$ for each $f$ in $C_mathcal{P}(X)$ is real valued. We show that each realcompact space lying between $X$ and $beta X$ is of the form $upsilon_{C_mathcal{P}}X$ if and only if $X$ is pseudocompact. We find out conditions under which an arbitrary product of spaces of the form locally-$mathcal{P}/$ almost locally-$mathcal{P}$, becomes a space of the same form. We further show that $C_mathcal{P}(X)$ is a free ideal ( essential ideal ) of $C(X)$ if and only if $C^mathcal{P}_infty(X)$ is a free ideal ( respectively essential ideal ) of $C^*(X)+C^mathcal{P}_infty(X)$ when and only when $X$ is locally-$mathcal{P}$ ( almost locally-$mathcal{P}$). We address the problem, when does $C_mathcal{P}(X)/C^mathcal{P}_{infty}(X)$ become identical to the socle of the ring $C(X)$. Finally we observe that the ideals of the form $C_mathcal{P}(X)$ of $C(X)$ are no other than the $z^circ$-ideals of $C(X)$.