  A proper ideal $I$ in a commutative ring with unity is called a
$z^\circ$-ideal if for each $a$ in $I$, the intersection of all minimal prime
ideals in $R$ which contain $a$ is contained in $I$. For any totally ordered
field $F$ and a completely $F$-regular topological space $X$, let $C(X,F)$ be
the ring of all $F$-valued continuous functions on $X$ and $B(X,F)$ the
aggregate of all those functions which are bounded over $X$. An explicit
formula for all the $z^\circ$-ideals in $A(X,F)$ in terms of ideals of closed
sets in $X$ is given. It turns out that an intermediate ring $A(X,F)\neq
C(X,F)$ is never regular in the sense of Von-Neumann. This property further
characterizes $C(X,F)$ amongst the intermediate rings within the class of
$P_F$-spaces $X$. It is also realized that $X$ is an almost $P_F$-space if and
only if each maximal ideal in $C(X,F)$ is $z^\circ$-ideal. Incidentally this
property also characterizes $C(X,F)$ amongst the intermediate rings within the
family of almost $P_F$-spaces.
