We further the study of rings with no middle class by focusing on an interpretation of that property in terms of the lattice of hereditary pretorsion classes over a given ring. For non-semisimple rings, the absence of a middle class is equivalent to the requirement that the class of all semisimple right modules be a coatom in that lattice. Taking advantage of this perspective, we discover new facts and shed light on others already known with a possibly more direct interpretation without having to refer to an exhaustive analysis of the structure theorems available in the literature. Our approach also allows us to characterize rings with no middle class in terms of hereditary pretorsion classes containing the class of all singular right modules. We discuss the open problem of whether there is a ring with no right middle class which is not right Noetherian and see, in particular, that an indecomposable ring satisfying that property would have to be Morita equivalent to a certain type of subring of a full linear ring.

中文翻译：

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从晶格理论的角度看没有中产阶级的戒指

我们通过以给定环上的遗传先天性类别的晶格为重点，对不具有中产阶级的环的研究进行了深入研究。对于非半简单的环，缺少中间类等于所有半简单右模块的类是该晶格中的复盖的要求。利用这种观点，我们发现了新的事实，并通过可能更直接的解释来阐明其他已知事实，而不必参考文献中可用的结构定理的详尽分析。我们的方法还允许我们根据包含所有奇异权模块的类的世袭类别来表征没有中产阶级的戒指。