本文選題：有限表現(xiàn)模 + 強表現(xiàn)模�。� 參考：《廣西師范大學(xué)》2015年碩士論文

【摘要】：本文主要分兩部分,第一部分定義并研究了一類特殊的幾乎有限表現(xiàn)模-幾乎強表現(xiàn)模.若M=M'(?)M*,其中M'是強表現(xiàn)模,M*是非有限生成自由模,則稱M是幾乎強表現(xiàn)的.我們給出了幾乎強表現(xiàn)模的一些等價刻畫,并得到幾乎強表現(xiàn)模在直和下保持封閉的性質(zhì).第二部分定義并研究了一類特殊的廣義有限表現(xiàn)模-廣義強表現(xiàn)模.M是R-模,若存在投射模P及強表現(xiàn)模A使得M(?)P/A,既有正合列0→A→P→M→0,則稱M為廣義強表現(xiàn)模.得到廣義強表現(xiàn)模的結(jié)構(gòu)定理,并討論了廣義強表現(xiàn)模的對偶模和廣義強表現(xiàn)模與幾乎強表現(xiàn)模的關(guān)系.本文具體內(nèi)容安排如下：第一章,概述幾乎強表現(xiàn)模與廣義強表現(xiàn)模的歷史背景和研究現(xiàn)狀,同時介紹了本文要用到的一些基本概念和常用符號.第二章,在幾乎有限表現(xiàn)模定義的基礎(chǔ)上定義了幾乎強表現(xiàn)模,給出了幾乎強表現(xiàn)模的一些等價刻畫,并探討幾乎強表現(xiàn)模的一些性質(zhì),證明了幾乎強表現(xiàn)模在直和下保持封閉.主要有以下結(jié)果：定理2.5設(shè)R是環(huán),M是R-模,則M是a.s.p.的充分必要條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.推論2.6設(shè)R是環(huán),M是R-模,則M是a.s.p.的充分必要條件是存在正合列0→F1→ F→M → 0,其中F1是強表現(xiàn)模,F是非有限生成自由模.定理2.7 設(shè)0→A→B→C→0為左R-模短正合列,若A,C均是a.s.p.的,則B也是a.s.p.的推論2.8 設(shè)R是環(huán),M1,M2,…,Mn是左R-模,若M1,M2,…,Mn均是a.s.p.模,則(?)i=1nMn也是a.s.p.的.命題2.12設(shè)R是環(huán),則對R上任意a.s.p.模M,有Fpd(M)=1第三章,在廣義有限表現(xiàn)模定義的基礎(chǔ)上定義了廣義強表現(xiàn)模,得到廣義強表現(xiàn)模的結(jié)構(gòu)定理,并討論了廣義強表現(xiàn)模的對偶模和廣義強表現(xiàn)模與幾乎強表現(xiàn)模的關(guān)系.主要有以下結(jié)果：定理3.1.3設(shè)R為環(huán),M是R-模,M是廣義強表現(xiàn)模的充分必要條件是存在投射模P0,自由模F*,強表現(xiàn)模Mo,使得M(?)P0=M0(?)F*定理3.2.2設(shè)R是一個環(huán)且R的每個投射模的有限生成子模都是強表現(xiàn)的,M是任意強表現(xiàn)模,則其對偶模M*=Hom(M,R)以及ExtRn(M,R)均是強表現(xiàn)模.定理3.2.3設(shè)R是一個環(huán)且R的每個投射模的有限生成子模都是強表現(xiàn)的,M是任意廣義強表現(xiàn)模,則ExtRn(M,R)是強表現(xiàn)的(n≥1).定理3.3.2設(shè)R為環(huán),M是R-模.M是廣義強表現(xiàn)模的充分必要條件是存在投射模P0,非有限生成自由模F*,強表現(xiàn)模Mo,使得M(?)P0=M0(?)F*推論3.3.3 設(shè)R為環(huán),M是左R-模,M為廣義強表現(xiàn)模當(dāng)且僅當(dāng)存在投射模P0使得M(?)P0是α.s.p.的.定理3.3.6設(shè)R是環(huán),則下列條件等價：(1)每個投謝R-模的強表現(xiàn)子模均為投射模,且任意強表現(xiàn)模是投射模；(2)任意a.s.p.左R-模是投射模；(3)對任意a.s.p.左R-模M,M的直和項是投射模；(4)對任意a.s.p.左R-模M,M的廣義強表現(xiàn)的直和項是投射模.
[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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