Rings Without a Middle Class from a Lattice-Theoretic Perspective

On the Splitting Property of Rings with Restricted Class of Injectivity Domains

“…We recall a useful lemma which is proved in [18] to produce portfolios from arbitrary hereditary pretorsion classes.…”

“…Observe that R is a ring with no middle class that is not semisimple Artinian if and only if SSMod-R is the unique coatom in the lattice iP(R). Rings with no middle class have been introduced in [1] and have been extensively studied in several papers (see, for example, [9,10,11,15,18]) from which we see that despite its quite simple profile, the structure of a ring with no middle class that is not semisimple Artinian can be too complicated. A key property for addressing challenges in the study of the structure of a ring R with no middle class is the existence of a ring decomposition R = S × T , where S is a semisimple Artinian ring and T is either zero or an indecomposable ring with no middle class such that the right socle of T is either zero or essential in T T .…”

“…To this end, we come up with the idea that the profile of the ring has exactly one coatom and call such profiles local. The method employed in this work is an extension of the techniques utilized in [18] and [19].…”

“…Since Y ′ is the unique coatom in the lattice of hereditary pretorsion classes of right T -modules by Theorem 2.7, T (M T ) ∈ T ∩ Mod-T ⊆ Y ′ ⊆ Y, and hence T (M ) ∈ Y. It follows that T ⊆ Y, completing the proof.The following corollary is now an immediate consequence of Theorem 2.11.Corollary 2.12 [18,. Theorem 3.8] Let R be a ring which is not semisimple Artinian.…”