Study of spectrum of certain subrings of a commutative ring with identity

TL;DR

This paper generalizes known topological results about spectra of certain subrings of continuous functions to arbitrary rings by introducing the concept of dense subrings, establishing homeomorphisms between their spectra.

Contribution

It introduces the notion of dense subrings in arbitrary rings and proves their spectra are homeomorphic under certain conditions, extending classical results.

Findings

Maximal spectra of dense subrings are homeomorphic.

Spectrum of a dense subring is densely embedded in the larger ring's spectrum.

Minimal prime ideals correspond uniquely between dense subrings and larger rings.

Abstract

By a ring we always mean a commutative ring with identity. It is well known that maximal spectrum of $C(X)$, $C^*(X)$ and any intermediate subrings between $C(X)$ and $C^* (X)$ are homeomorphic and homeomorphic with $\beta X$, the Stone-$\check{C}$ech compactification of $X$. In this paper we generalized these results to an arbitrary ring by introducing a notion of dense subring. We proved that if $A$ is completely normal and dense subring of $B$, then maximal spectrum of $A$ and $B$ are homeomorphic and hence maximal spectrum of all intermediate subrings between $A$ and $B$ where $A$ is dense, are homeomorphic. We also proved that $A$ is dense subring of $B$ if and only if spectrum of $B$ is densely embedded in spectrum of $A$ and have further shown that if $A$ is dense subring of $B$, any minimal prime ideal of $A$ is precisely of the form $Q\cap A$ for some unique minimal prime ideal $Q$ of $B$. As a consequence, we concluded that if $A$ is dense in $B$, minimal spectrum of $A$ and that of $B$ are homeomorphic. We also studied different properties of dense subrings of a ring.