Abundance of Isomorphic and non isomorphic intermediate rings

TL;DR

This paper investigates the diversity of intermediate rings within the ring of continuous functions on certain topological spaces, establishing the existence of many non-isomorphic rings under specific conditions.

Contribution

It demonstrates that for first countable, real compact spaces, the family of intermediate rings contains at least 2^c non-isomorphic rings, extending known results.

Findings

At least 2^c many distinct intermediate rings for non-pseudocompact spaces.

In first countable, real compact spaces, these rings are pairwise non-isomorphic.

The result generalizes previous knowledge about the structure of intermediate rings.

Abstract

It is well known that for a non pseudocompact space X, the family (X) of all intermediate subrings of C(X) which contain bounded real valued continuous functions contains at least 2c many distinct rings. We show that if in addition X is first countable and real compact, then there are at least 2c many rings in (X), no two of which are pairwise isomorphic.