On Artinian Rings with Restricted Class of Injectivity Domains

TL;DR

This paper characterizes Artinian rings with no middle class, describing their structure and providing full classifications, especially for commutative and non-simple cases, advancing the understanding of injectivity domain restrictions.

Contribution

It offers a comprehensive structural and classification framework for Artinian rings with no middle class, including new characterizations for commutative and non-simple rings.

Findings

Characterization of right Artinian right nonsingular rings with no right middle class.

Full classification of rings with no middle class.

Description of commutative rings with no middle class as sums of semisimple and length-two rings.

Abstract

In a recent paper of Alahmadi, Alkan and Lopez-Permouth, a ring R is defined to have no (simple) middle class if the injectivity domain of any (simple) R-module is the smallest or largest possible. Er, Lopez-Permouth and Sokmez use this idea of restricting the class of injectivity domains to classify rings, and give a partial characterization of rings with no middle class. In this work, we continue the study of the property of having no (simple) middle class. We give a structural description of right Artinian right nonsingular rings with no right middle class. We also give a characterization of right Artinian rings that are not SI to have no middle class, which gives rise to a full characterization of rings with no middle class. Furthermore, we show that commutative rings with no middle class are those Artinian rings which decompose into a sum of a semisimple ring and a ring of composition length two. Also, Artinian rings with no simple middle class are characterized. We demonstrate our results with several examples.