A note on rings of finite rank

TL;DR

This paper explores various constructions of rings with a specified finite rank, extending previous results and providing explicit examples across different dimensions and ring types.

Contribution

It introduces four new constructions of rings with any given finite rank, including generalizations of existing methods and novel examples involving Artinian and polynomial rings.

Findings

Every positive integer can be realized as the rank of some ring.

Multiple constructions exist for rings of finite rank in dimensions 0 and 1.

Explicit examples include one-dimensional domains and polynomial rings over Artinian rings.

Abstract

The rank of a ring $R$ is the supremum of minimal cardinalities of generating sets of $I$ as $I$ ranges over ideals of $R$. Matson showed that every positive integer occurs as the rank of some ring $R$. Motivated by the result of Cohen and Gilmer that a ring of finite rank has Krull dimension $0$ or $1$, we give four different constructions of rings of rank $n$ (for all positive integers n). Two constructions use one-dimensional domains, and the former of these directly generalizes Matson's construction. Our third construction uses Artinian rings (dimension zero), and our last construction uses polynomial rings over local Artinian rings (dimension one, irreducible, not a domain).