Restricted minimum condition in reduced commutative rings

TL;DR

This paper explores the properties of restricted minimum (RM) rings, especially in reduced Noetherian rings, generalizing classical results, and providing new characterizations and examples across various classes of rings.

Contribution

It extends the understanding of RM rings beyond Noetherian domains to reduced Noetherian rings, offering new characterizations and examples, including polynomial rings and Dedekind domains.

Findings

RM rings in Noetherian domains have Krull dimension at most one

Affine rings of curves are RM

Polynomial ring R[x] is RM iff R is reduced Artinian

Abstract

We say that a commutative ring R satisfies the restricted minimum (RM) condition if for all essential ideals I in R, factor R/I is an Artinian ring. We will focus on Noetherian reduced rings because in this setting known results for RM domains generalize well. However, as we will show, RM rings need not be Noetherian and may have nilpotent elements. One of the classic results in the theory of RM rings is that for Noetherian domains RM condition corresponds to having Krull dimension at most one. We will show that this can be generalized to reduced Noetherian rings, thus proving that affine rings corresponding to curves are RM. We will give examples showing that the assumption that the ring is reduced is not superfluous. We will prove that CDR domains are RM and this will allow us to give a new characterization of Dedekind domains. Examples of RM rings for various classes of rings will be given. In particular, we will show that a ring of polynomials R[x] is RM if and only if R is reduced Artinian ring. And we will study the relation between RM rings and UFDs.