  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.
