On Artinian rings with restricted class of injectivity domains

In a recent paper of Alahmadi, Alkan and López-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, López-Permouth and Sökmez 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.