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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 primeideals in $R$ which contain $a$ is contained in $I$. For any totally orderedfield $F$ and a completely $F$-regular topological space $X$, let $C(X,F)$ bethe ring of all $F$-valued continuous functions on $X$ and $B(X,F)$ theaggregate of all those functions which are bounded over $X$. An explicitformula for all the $z^\circ$-ideals in $A(X,F)$ in terms of ideals of closedsets in $X$ is given. It turns out that an intermediate ring $A(X,F)\neqC(X,F)$ is never regular in the sense of Von-Neumann. This property furthercharacterizes $C(X,F)$ 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 andonly if each maximal ideal in $C(X,F)$ is $z^\circ$-ideal. Incidentally thisproperty also characterizes $C(X,F)$ amongst the intermediate rings within thefamily of almost $P_F$-spaces.

$z^\circ$-ideals in intermediate rings of ordered field valued continuous functions