Topics in torsion theory
Topics in torsion theory
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DOI：
10.1017/s0004972700039071
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发表时间：
2007-02
影响因子：
0.7
通讯作者：
Stelios A. Charalambides
中科院分区：
文献类型：
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作者：
Stelios A. Charalambides
本文的目的是将环和模理论中的各种概念和结果推广到扭转理论的集合中。为了做到这一点，我们首先做一些初步的介绍，介绍了在扭转理论中使用的主要思想，主要是-扭转和-无扭转模块以及-密集和-纯子模块。在第一章中，我们还介绍了一个新的概念，即-紧模块，它是足够基本的，应该在入门中占有一席之地。我们得到的结果分为三个方面，它们在一定程度上是相互关联的。第一个区域是-Max模，我们将其作为Max模的扭转理论模拟引入。主要目的是推广激波的一个众所周知的结果，该结果是通过使用社会，根型和最大模来表征noether环。所有这些概念在我们的推广中都有对应的扭转理论。此外，在最近一篇引入短模的文章的启发下，我们定义和刻画了左最大环，并应用激波定理的扭转理论版本得到了短模的刻画。第二个领域涉及各种注入性，有些是已知的，有些是新的。我们引入了-M -内射和s-M -内射模，并研究了它们与已知的-内射和-拟内射模的关系。在此基础上，给出了表征s-M -内射模的广义Fuchs准则的改进版本，并对Azumaya引理进行了推广。我们还证明了每个M生成的模都有一个-M -内射壳，该壳在同构上是唯一的，并说明了它是如何与-拟内射壳联系起来的。然后我们研究Σ-injectivity，推广了Faith， Albu和nast asescu和Cailleau的著名结果，这些结果为Σinjective性质、Σ-s-M -内射性质和Σ-s-M -内射模块的直接和为Σ-s-M -内射提供了必要和辅助条件。在第三个领域，我们引入了几个新概念，目的是将CS或扩展模块的概念引入扭转理论设置。这种方法是双重的。第一个是通过-CS模块，它是CS模块以及-拟连续、-拟内射和-内射模块的推广；第二个是通过s-CS模块，它是CS模块的一种特殊情况。我们的动机是提供一个著名的结果的扭转理论模拟由Okado，表征诺etherian模块。我们使用s-CS模块和一个很好的扭转理论模拟得到了一些部分结果，尽管没有使用-CS或s-CS模块。我们还检验了我们的相关版本的CS模块与其他作者的版本之间的关系，并获得了他们的一些结果的补充。
The purpose of this thesis is to generalize to the torsion-theoretic setting various concepts and results from the theory of rings and modules. In order to accomplish this we begin with some preliminaries which introduce the main ideas used in torsion theory, the major ones being -torsion and -torsionfree modules as well as -dense and -pure submodules. In the rst chapter we also introduce a new concept, that of a -compact module, which is basic enough to deserve a place among the preliminaries. The results that we obtain fall into three areas which are to a certain degree interrelated. The rst area is on -Max modules, which we introduce as a torsion-theoretic analogue of Max modules. The main aim is to generalize a well-known result by Shock which characterizes Noetherian rings by using the socle, the radical and Max modules. All of these concepts have torsiontheoretic counterparts which we utilize in our generalization. Furthermore, we de ne and characterize left -Max rings and apply the torsion-theoretic version of Shock's theorem to obtain a characterization of -short modules motivated by a recent article in which short modules were introduced. The second area deals with various avours of -injectivity, some known and some new. We introduce -M -injective and s-M -injective modules and examine their relationship with the known concepts of -injective and -quasiinjective modules. We then provide an improved version of the Generalized Fuchs Criterion which characterizes s-M -injective modules, and give a generalization of Azumaya's Lemma. We also prove that every M -generated module has a -M -injective hull which is unique up to isomorphism and show how this is linked to the -quasi-injective hull. We then examine Σ-injectivity, generalizing well-known results by Faith, Albu and N ast asescu and Cailleau which provide necessary and su cient conditions for the Σinjective property, the Σ-s-M -injective property and for a direct sum of Σ-s-M -injective modules to be Σ-s-M -injective. In the third area we introduce a couple of new concepts with the aim of bringing to the torsion-theoretic setting the concept of a CS or extending module. The approach is twofold. The rst is via -CS modules which serve as a generalization of CS modules as well as -quasi-continuous, -quasi-injective and -injective modules, and the second is via s-CS modules which are a special case of CS modules. Our motivation is to provide a torsion-theoretic analogue of a well-known result by Okado which characterizes Noetherian modules. We have some partial results using s-CS modules and a nice torsion-theoretic analogue, albeit without the use of -CS or s-CS modules. We also examine the relationship between our relative versions of CS modules with those of other authors and obtain re nements to some of their results.