Poor modules with no proper poor direct summands

Ponente(s): Sergio Lopez-permouth ., Rafail Alizade, Engin Buyukasik, y Liu Yang.

As a mean to provide intrinsic characterizations of poor modules, the notion of a pauper module is introduced. A module is a pauper if it is poor and has no proper poor direct summand. We show that not all rings have pauper modules and explore conditions on rings for the existence of such modules. In addition, we ponder the role of paupers in the characterization of poor modules over those rings that do have them. It is shown that the existence of paupers is equivalent to the Noetherian condition for rings with no middle class. As indecomposable poor modules are pauper, we study rings with no indecomposable right middle class (i.e. the ring whose indecomposable right modules are pauper or injective) and show that a commutative Noetherian ring R has no indecomposable middle class if and only if R = S X T, with S semisimple Artinian and T a local ring whose unique maximal ideal is minimal. The structure of poor modules is completely determined over commutative hereditary Noetherian rings. Pauper Abelian groups with torsion-free rank one are fully characterized.