Quasi-maximal ideals and ring extensions

TL;DR

This work develops a comprehensive framework for quasi-maximal and submaximal ideals in extensions of commutative rings. It characterizes quasi-maximal ideals through multiple equivalences, connects them to submaximal ideals, and analyzes their behavior under ring maps and in minimal and finite extensions. A key highlight is the conductor of a finite minimal extension being quasi-maximal in the target ring, which yields a new lens to classify and understand these extensions. The paper also clarifies the relationship between quasi-maximal and Badawi’s 2-absorbing ideals, including conditions under which these notions coincide, and provides Appendix tools on t-closures and saturation that illuminate the local-global structure of quasi-maximal ideals. Overall, the results offer precise transfer principles across extensions and contribute to the structural theory of zero-dimensional and Artinian quotients arising from quasi-maximal ideals.

Abstract

Alan and al. defined and studied quasi-maximal ideals. We add a comprehensive characterization of these ideals, introducing submaximal ideals. The conductor of a finite minimal extension $R\subset S$ is quasi-maximal in $S$. This allows us to give a new characterization of these extensions. We also examine the links between quasi-maximal ideals and Badawi 2-absorbing ideals.

Theorems & Definitions (175)

• Proposition 2.1
• proof
• Proposition 2.2
• Definition 2.3
• Theorem 2.4
• proof
• Proposition 2.5
• proof
• Remark 2.6
• Definition 2.7