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Research on Some Problems of Indecomposable Modules

Author: XuWei
Tutor: FengLiangGui
School: National University of Defense Science and Technology
Course: Basic mathematics
Keywords: Indecomposable Module Uniform Module Generating Set Indecomposable Abelian Group T1 ring
CLC: O153.3
Type: Master's thesis
Year: 2009
Downloads: 21
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Abstract


The problems about indecomposable modules are researching focus in the field of module theory. The indecomposable module is the generalized conception of simple module. So the researching about the classification of modules is usually turned into the study of the indecomposable modules. In this thesis, we will talk about some of our researching work on three problems about indecomposable modules, which are the problems about construction properties of indecomposable modules, the structure of indecomposable modules and the characterization of rings by the indecomposable modules over it.For general ring, the construction of the indecomposable modules are very complex. This can be seen from the related results in papers [3], [4] and[5]. In the second charpter of this thesis, for a class of special rings, we get a main result on the construction property of indecomposable modules (Theorem 2.2.4) . Particularly, this class of rings contains the regular local domain of demision 1. Using this main result, in the forth chapter we will do some further research about the classification of indecomposable Abelian groups, which are the indecomposable modules over the domain Z. In that section, we will get a result on the classification of indecomposable Abelian groups. By this classifyed theorem we construct some examples of indecomposable Abelian groups, which can be used to illustrate some properties of indecomposable Abelian groups.It is known that the indecomposable modules can be used to charaterize the properties of the ring. Examples are the results of F. Couchot (the article [11]). In the third chapter, we do some more research about this problem. By comparison between the indecomposable modules and uniform modules, we define a new class of rings, named T1 ring, and prove some results about the properties of this class of rings. Furthermore, we prove that the T1 ring is the proper extended conception of semi-simple ring and give some examples of T1 ring which are not semi-simple. It could be belived that the T1 ring is the ring which is more general than semi-simple ring, but the indecomposable modules over it still have nice properties.

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CLC: > Mathematical sciences and chemical > Mathematics > Algebra,number theory, portfolio theory > Abstract algebra ( Algebra ) > Ring Theory
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