Resumen de A class of ideals in intermediate rings of continuous functions

Sagarmoy Bag, Sudip Kumar Acharyya, Dhananjoy Mandal

• For any completely regular Hausdorff topological space $X$, an intermediate ring $A(X)$ of continuous functions stands for any ring lying between $C^*(X)$ and $C(X)$. It is a rather recently established fact that if $A(X)\neq C(X)$, then there exist non maximal prime ideals in $A(X)$. We offer an alternative proof of it on using the notion of $z^\circ$ ideals. It is realized that a $P$-space $X$ is discrete if and only if $C(X)$ is identical to the ring of real valued measurable functions defined on the $\sigma $-algebra $\beta (X)$ of all Borel sets in $X$. Interrelation between three classes of ideals viz $z$-ideals, $z^\circ$-ideal and $\mathfrak{Z}_A$-ideals in $A(X)$ are examined. It is proved that with in the family of almost $P$-spaces $X$, each $\mathfrak{Z}_A$-ideal in $A(X)$ is a $z^\circ $-ideal if and only if each $z$-ideal in $A(X)$ is a $z^\circ$-ideal if and only if $A(X)=C(X)$.