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

TL;DR

This paper characterizes $z^\u00b0$-ideals in intermediate rings of continuous functions valued in ordered fields, providing explicit formulas and exploring their properties in relation to space types like $P_F$-spaces.

Contribution

It offers an explicit formula for all $z^\u00b0$-ideals in intermediate rings of ordered field-valued continuous functions and characterizes the ring $C(X,F)$ among intermediate rings.

Findings

Intermediate rings are never regular in the Von-Neumann sense.

$C(X,F)$ is characterized among intermediate rings by properties of $P_F$-spaces.

Maximal ideals in $C(X,F)$ are $z^\u00b0$-ideals if and only if $X$ is an almost $P_F$-space.

Abstract

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.