An infinite cardinal-valued Krull dimension for rings

TL;DR

This paper introduces two generalized notions of Krull dimension for rings that can take any cardinal value, and explores the existence of rings with specified cardinalities and dimensions under various set-theoretic assumptions.

Contribution

It defines the concepts of cardinal Krull dimension and strong cardinal Krull dimension, and characterizes when rings with given cardinalities and dimensions exist, including constructions under different set-theoretic assumptions.

Findings

Complete characterization for K ≥ L cases.

Constructed rings with K < L using valuation rings, polynomial rings, and Leavitt path algebras.

Dependent on set-theoretic assumptions for certain cardinal pairs.

Abstract

We define and study two generalizations of the Krull dimension for rings, which can assume cardinal number values of arbitrary size. The first, which we call the "cardinal Krull dimension," is the supremum of the cardinalities of chains of prime ideals in the ring. The second, which we call the "strong cardinal Krull dimension," is a slight strengthening of the first. Our main objective is to address the following question: for which cardinal pairs (K,L) does there exist a ring of cardinality K and (strong) cardinal Krull dimension L? Relying on results from the literature, we answer this question completely in the case where K>L or K=L. We also give several constructions, utilizing valuation rings, polynomial rings, and Leavitt path algebras, of rings having cardinality K and (strong) cardinal Krull dimension L>K. The exact values of K and L that occur in this situation depend on set-theoretic assumptions.