On Homology Characterization Of Pure Module

A lot of articles have characterized the purity of the module since 1960's, in which the concepts were introduced. We can refer to [1]-[6]. Factor module is one of the most important tools to study modules and rings. In this paper, the author use factor modules to characterize the pure modules.In the first chapter, we introduce some elementary definitions and important results, which are the useful basis of the following study.In the second chapter, some special factor modules of the pure modules are studied. Firstly, we study the modules which factor modules are flat. We get that a ring R is V.N. regular ring if and only if every factor module of the injective .R-module is flat(refer to theory 2.1.9). The result improves the important claim that a ring R is V.N. regular ring if and only if every .R-module is flat.Secondly, we check the pure simple modules with the factor modules, and introduce P.RadM (P.RadM = ∩{K ≤ M|M/K is pure semisimple modules}), then obtained equivalent characterization on P.RadM=0. And therefore, it enrich the character of pure-semisimple modules and rings.Finally, factor modules are used to character the pure dimension, and we focus on the rings which pure global dimension less than 1 or 2.In the third chapter, we study some special pure modules. In the first part, we study the ring which simple modules are pure injective, and obtain the characterization of pure semisimple ring(refer to theory 3.1.8). In the other part, we study the rings in which every closed submodule of finite generated flat module is pure, then characterize this ring over π-coherent ring. Meanwhile, we characterize the well-known rings, such as quasi-perfect rings and hereditary rings with abovementioned ring.