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Modules satisfying conditions that are opposites of absolute purity and flatness
(Izmir Institute of Technology, 2017-07) Kafkas Demirci, Gizem; Büyükaşık, Engin
The main purpose of this thesis is to study the properties which are opposites of absolute pure and flat modules. A right module M is said to be test for flatness by subpurity (for short, t.f.b.s.) if its subpurity domain is as small as possible, namely, consisting of exactly the flat left modules. A left module M is said to be rugged if its flatness domain is the class of all regular right R-modules. Every ring has a t.f.b.s. module. For a right Noetherian ring R every simple right R-module is t.f.b.s. or absolutely pure if and only if R is a right V-ring or R A×B, where A is right Artinian with a unique non-injective simple right R-module and Soc(AA) is homogeneous and B is semisimple. A characterization of t.f.b.s. modules over commutative hereditary Noetherian rings is given. Rings all (cyclic) modules of whose are rugged are shown to be von Neumann regular rings. Over a right Noetherian ring every left module is rugged or flat if and only if every right module is poor or injective if and only if R = S × T, where S is semisimple Artinian and T is either Morita equivalent to a right PCI-domain, or T is right Artinian whose Jacobson radical contains no properly nonzero ideals. Connections between rugged and poor modules are shown. Rugged Abelian groups are fully characterized.