Almost clean rings and arithmetical rings

TL;DR

This paper characterizes when certain commutative rings are elementary divisor rings using topological properties of their prime spectra, introduces concepts like almost clean rings, and explores their structural implications.

Contribution

It provides new criteria for elementary divisor rings based on the topology of prime spectra and studies properties of almost clean rings and their quotients.

Findings

A commutative Bézout ring with compact minimal prime spectrum is an elementary divisor ring if and only if all its quotients by minimal primes are.

In arithmetical rings, the quotient space of prime spectrum is Hausdorff and homeomorphic to the minimal prime spectrum when compact.

Almost clean rings have totally disconnected prime spectrum and their quotients are also almost clean, with converse under finitely presented principal ideals.

Abstract

It is shown that a commutative B\'ezout ring $R$ with compact minimal prime spectrum is an elementary divisor ring if and only if so is $R/L$ for each minimal prime ideal $L$. This result is obtained by using the quotient space $\mathrm{pSpec} R$ of the prime spectrum of the ring $R$ modulo the equivalence generated by the inclusion. When every prime ideal contains only one minimal prime, for instance if $R$ is arithmetical, $\mathrm{pSpec} R$ is Hausdorff and there is a bijection between this quotient space and the minimal prime spectrum $\mathrm{Min} R$, which is a homeomorphism if and only if $\mathrm{Min} R$ is compact. If $x$ is a closed point of $\mathrm{pSpec} R$, there is a pure ideal $A(x)$ such that $x=V(A(x))$. If $R$ is almost clean, i.e. each element is the sum of a regular element with an idempotent, it is shown that $\mathrm{pSpec} R$ is totally disconnected and, $\forall x\in\mathrm{pSpec} R$, $R/A(x)$ is almost clean; the converse holds if every principal ideal is finitely presented. Some questions posed by Facchini and Faith at the second International Fez Conference on Commutative Ring Theory in 1995, are also investigated. If $R$ is a commutative ring for which the ring $Q(R/A)$ of quotients of $R/A$ is an IF-ring for each proper ideal $A$, it is proved that $R_P$ is a strongly discrete valuation ring for each maximal ideal $P$ and $R/A$ is semicoherent for each proper ideal $A$.