几乎强表现模与广义强表现模

发布时间：2018-07-12 13:31

  本文选题：有限表现模 + 强表现模　； 参考：《广西师范大学》2015年硕士论文

【摘要】：本文主要分两部分,第一部分定义并研究了一类特殊的几乎有限表现模-几乎强表现模.若M=M'(?)M*,其中M'是强表现模,M*是非有限生成自由模,则称M是几乎强表现的.我们给出了几乎强表现模的一些等价刻画,并得到几乎强表现模在直和下保持封闭的性质.第二部分定义并研究了一类特殊的广义有限表现模-广义强表现模.M是R-模,若存在投射模P及强表现模A使得M(?)P/A,既有正合列0→A→P→M→0,则称M为广义强表现模.得到广义强表现模的结构定理,并讨论了广义强表现模的对偶模和广义强表现模与几乎强表现模的关系.本文具体内容安排如下：第一章,概述几乎强表现模与广义强表现模的历史背景和研究现状,同时介绍了本文要用到的一些基本概念和常用符号.第二章,在几乎有限表现模定义的基础上定义了几乎强表现模,给出了几乎强表现模的一些等价刻画,并探讨几乎强表现模的一些性质,证明了几乎强表现模在直和下保持封闭.主要有以下结果：定理2.5设R是环,M是R-模,则M是a.s.p.的充分必要条件是存在正合列其中F0是非有限生成自由模,Fi(i=1,2,…m,…)是有限生成投射模.推论2.6设R是环,M是R-模,则M是a.s.p.的充分必要条件是存在正合列0→F1→ F→M → 0,其中F1是强表现模,F是非有限生成自由模.定理2.7 设0→A→B→C→0为左R-模短正合列,若A,C均是a.s.p.的,则B也是a.s.p.的推论2.8 设R是环,M1,M2,…,Mn是左R-模,若M1,M2,…,Mn均是a.s.p.模,则(?)i=1nMn也是a.s.p.的.命题2.12设R是环,则对R上任意a.s.p.模M,有Fpd(M)=1第三章,在广义有限表现模定义的基础上定义了广义强表现模,得到广义强表现模的结构定理,并讨论了广义强表现模的对偶模和广义强表现模与几乎强表现模的关系.主要有以下结果：定理3.1.3设R为环,M是R-模,M是广义强表现模的充分必要条件是存在投射模P0,自由模F*,强表现模Mo,使得M(?)P0=M0(?)F*定理3.2.2设R是一个环且R的每个投射模的有限生成子模都是强表现的,M是任意强表现模,则其对偶模M*=Hom(M,R)以及ExtRn(M,R)均是强表现模.定理3.2.3设R是一个环且R的每个投射模的有限生成子模都是强表现的,M是任意广义强表现模,则ExtRn(M,R)是强表现的(n≥1).定理3.3.2设R为环,M是R-模.M是广义强表现模的充分必要条件是存在投射模P0,非有限生成自由模F*,强表现模Mo,使得M(?)P0=M0(?)F*推论3.3.3 设R为环,M是左R-模,M为广义强表现模当且仅当存在投射模P0使得M(?)P0是α.s.p.的.定理3.3.6设R是环,则下列条件等价：(1)每个投谢R-模的强表现子模均为投射模,且任意强表现模是投射模；(2)任意a.s.p.左R-模是投射模；(3)对任意a.s.p.左R-模M,M的直和项是投射模；(4)对任意a.s.p.左R-模M,M的广义强表现的直和项是投射模.
[Abstract]:This paper is divided into two parts. In the first part, we define and study a special class of almost finite representation modules-almost strong representation modules. If M'is a strong representation module and M * is a nonfinitely generated free module, then M is almost strong. We give some equivalent characterizations of almost strong representation modules and obtain the property that almost strong representation modules remain closed under the direct sum. In the second part, we define and study a class of special generalized finite representation modules. M is a R- module. If there exists a projective module P and a strong representation module A such that M (?) P / A, both positive sequences 0 A and P + M 0, then M is called generalized strong representation module. The structure theorems of generalized strongly expressive modules are obtained, and the relations between dual modules, generalized strong representation modules and almost strong representation modules of generalized strong representation modules are discussed. The main contents of this paper are as follows: the first chapter summarizes the historical background and research status of almost strong representation module and generalized strong representation module, and introduces some basic concepts and common symbols used in this paper. In chapter 2, on the basis of the definition of almost finite representation module, we define almost strong representation module, give some equivalent characterizations of almost strong representation module, and discuss some properties of almost strong representation module. It is proved that almost strong representation modules remain closed under the direct sum. The main results are as follows: theorem 2.5 Let R be a ring M is a R-module, then M is a.s.p. If and only if there is an exact sequence where F _ 0 is a non-finite-generated free module F _ I (I ~ (1) F ~ (2). M,.) Is a finitely generated projective module. Corollary 2.6 Let R be a ring M is a R-module, then M is a.s.p. A necessary and sufficient condition is that there exists an exact sequence of 0 F _ 1 F ~ F ~ F ~ M ~ 0, where F _ 1 is a strong representation module F _ F is a non-finitely generated free module. Theorem 2.7 Let 0 A ~ (B) C ~ (0) be a short positive sequence of left R- modules, if AZC are all A. s. P. B is also a.s.p. Let R be a ring M _ (1) M _ (1) M _ (2),. Mn is a left R- module, if M1, M2,. The content of mn is a.s.p. Module, then (?) iP1nMn is also a.s.p. Of. Proposition 2.12 Let R be a ring, then for any a.s.p. on R, let R be a ring. In chapter 3, the generalized strong representation module is defined on the basis of the definition of the generalized finite representation module, and the structure theorem of the generalized strong representation module is obtained. The relations between the dual module and the generalized strong representation module and the almost strong representation module of the generalized strong representation module are discussed. The main results are as follows: theorem 3.1.3 Let R be a ring M is a R -module M is a generalized strong representation module if and only if there is projective module P0, free module FG, strong representation module Mo. such that M (?) P0P0 M0 (?) F * theorem 3.2.2 Let R be a ring and every projective module of R The finitely generated submodules are strongly represented and M is an arbitrary strong representation module. Then the dual modules Mannon Hom (Mnr) and ExtRn (MKR) are all strong phenomodules. Theorem 3.2.3 Let R be a ring and the finite generated submodules of every projective module of R are strongly represented and M is an arbitrary generalized strongly represented module, then ExtRn (MannR) is strongly expressed (n 鈮，

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