Noetherian rings of non-local rank

TL;DR

This paper characterizes Noetherian rings whose global rank differs from the supremum of local ranks, showing they decompose into specific local rings, and applies this to compute polynomial ring ranks over Artinian rings.

Contribution

It provides a new characterization of certain Noetherian rings based on their rank and localizations, revealing their decomposition into principal Artinian rings and Dedekind domains.

Findings

Rings with rank not equal to the supremum of local ranks are finite products of local principal Artinian rings and Dedekind domains.

Such rings necessarily decompose into specific local components, at least one not being a principal ideal ring.

The rank of polynomial rings over Artinian rings can be computed using local properties.

Abstract

The rank of a ring $R$ is the supremum of minimal cardinalities of generating sets of $I$, among all ideals $I$ in $R$. In this paper, we obtain a characterization of Noetherian rings $R$ whose rank is not equal to the supremum of ranks of localizations of $R$ at maximal ideals. It turns out that any such ring is a direct product of a finite number of local principal Artinian rings and Dedekind domains, at least one of which is not a principal ideal ring. As an application, we show that the rank of the ring of polynomials over an Artinian ring can be computed locally.