Module Theory: Endomorphism rings and direct sum decompositions in some classes of modules
Module Theory: Endomorphism rings and direct sum decompositions in some classes of modules
复制标题
模块理论：某些类模块中的自同态环和直和分解
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发表时间：
1998
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通讯作者：
A. Facchini
中科院分区：
文献类型：
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作者：
A. Facchini
1基本概念。- 1.1半简单环和模。- 1.2局部环和半局部环。—1.3串口环和模块。- 1.4纯精确序列。- 1.5有限可定义的子群和纯内射模。- 1.6类别（RFP, Ab）。- 1.7 ？-pure-injective模块。- 1.8关于第一章的注释。- 2 Krull-Schmidt-Remak-Azumaya定理。- 2.1交换属性。- 2.2带有exchange属性的不可分解模块。- 2.3有限直接和分解的同构改进。- 2.4 Krull-Schmidt-Remak-Azumaya定理。—2.5应用。- 2.6模格的戈尔迪维数。- 2.7模块的金色尺寸。- 2.8模块的双Goldie尺寸。- 2.9 ？-小模块和？封闭类。- 2.10直接金额为？小模块。- 2.11 Loewy系列。- 2.12交换环或右noether环上的Artinian右模。- 2.13关于第二章的说明。- 3枚半完美戒指。- 3.1投射盖和提升幂等。- 3.2半完美环。- 3.3半完全环上的模。- 3.4有限呈现和拟合模块。- 3.5串行环上有限呈现的模块。- 3.6关于第三章的注释。- 4个半局部环。- 4.1坎普-迪克斯定理。- 4.2具有半局部自同态环的模。—4.3举例。- 4.4关于第4章的注释。- 5个系列戒指。—5.1链环和右链环。- 5.2 artinian系列环上的模块。- 5.3非奇异和半遗传的序列环。- 5.4诺瑟系列环。- 5.5第五章注释- 6个商环。- 6.1任意环的商环。- 6.2右Goldie环无子环。6.3等级降低。- 6.4链环的定位。- 6.5串行环中的可定位系统。- 6.6举例。- 6.7串环的素数理想。- 6.8 Goldie半素数理想。- 6.9矩阵的对角化。- 6.10矿石套在串行环中。- 6.11 Goldie半素数理想和极大Ore集。- 6.12串环的经典商环。- 6.13第六章注释。- 7克鲁尔尺寸和系列环。- 7.1标尺偏差。- 7.2任意模和环的库尔维数。- 7.3右Krull维环的零子环。- 7.4雅各布森基的超幂。- 7.5有限克鲁尔维数串联环的结构。- 7.6关于第七章的注释。Krull-Schmidt在有限生成模块和Artinian模块中失败。- 8.1 Krull-Schmidt有限生成模块失败。- 8.2 Krull-Schmidt模块失效- 8.3第8章注释。- 9双均匀模块。- 9.1双均匀模块的第一属性。9.2一些技术引理。- 9.3充分条件。- 9.4双均匀模的弱Krull-Schmidt定理。9.5 Krull-Schmidt定理适用于链环上有限呈现的模。- 9.6 Krull-Schmidt在串行环上有限呈现模块失败。- 9.7类型1的双均匀模块的进一步示例。—9.8准小串行模块。- 9.9串行模块族的必要条件。- 9.10第九章注释。- 10 ？-纯注入模和Artinian模。- 10.1枚忠实的戒指？-pure-injective模块。- 10.2与人工模同构环同构的环。—10.3分布式模块。- 10.4 ？-环上的纯注入模。- 10.5同质？-pure-injective模块。- 10.6克鲁尔维度和？-pure-injective模块。- 10.7为单模自同态环的序列环。- 10.8可本地化的系统和？-序列环上的纯内射模。- 10.9关于第十章的注释。- 11个开放性问题。
1 Basic Concepts.- 1.1 Semisimple rings and modules.- 1.2 Local and semilocal rings.- 1.3 Serial rings and modules.- 1.4 Pure exact sequences.- 1.5 Finitely definable subgroups and pure-injective modules.- 1.6 The category (RFP, Ab).- 1.7 ?-pure-injective modules.- 1.8 Notes on Chapter 1.- 2 The Krull-Schmidt-Remak-Azumaya Theorem.- 2.1 The exchange property.- 2.2 Indecomposable modules with the exchange property.- 2.3 Isomorphic refinements of finite direct sum decompositions.- 2.4 The Krull-Schmidt-Remak-Azumaya Theorem.- 2.5 Applications.- 2.6 Goldie dimension of a modular lattice.- 2.7 Goldie dimension of a module.- 2.8 Dual Goldie dimension of a module.- 2.9 ?-small modules and ?-closed classes.- 2.10 Direct sums of ?-small modules.- 2.11 The Loewy series.- 2.12 Artinian right modules over commutative or right noetherian rings.- 2.13 Notes on Chapter 2.- 3 Semiperfect Rings.- 3.1 Projective covers and lifting idempotents.- 3.2 Semiperfect rings.- 3.3 Modules over semiperfect rings.- 3.4 Finitely presented and Fitting modules.- 3.5 Finitely presented modules over serial rings.- 3.6 Notes on Chapter 3.- 4 Semilocal Rings.- 4.1 The Camps-Dicks Theorem.- 4.2 Modules with semilocal endomorphism ring.- 4.3 Examples.- 4.4 Notes on Chapter 4.- 5 Serial Rings.- 5.1 Chain rings and right chain rings.- 5.2 Modules over artinian serial rings.- 5.3 Nonsingular and semihereditary serial rings.- 5.4 Noetherian serial rings.- 5.5 Notes on Chapter 5.- 6 Quotient Rings.- 6.1 Quotient rings of arbitrary rings.- 6.2 Nil subrings of right Goldie rings.- 6.3 Reduced rank.- 6.4 Localization in chain rings.- 6.5 Localizable systems in a serial ring.- 6.6 An example.- 6.7 Prime ideals in serial rings.- 6.8 Goldie semiprime ideals.- 6.9 Diagonalization of matrices.- 6.10 Ore sets in serial rings.- 6.11 Goldie semiprime ideals and maximal Ore sets.- 6.12 Classical quotient ring of a serial ring.- 6.13 Notes on Chapter 6.- 7 Krull Dimension and Serial Rings.- 7.1 Deviation of a poset.- 7.2 Krull dimension of arbitrary modules and rings.- 7.3 Nil subrings of rings with right Krull dimension.- 7.4 Transfinite powers of the Jacobson radical.- 7.5 Structure of serial rings of finite Krull dimension.- 7.6 Notes on Chapter 7.- 8 Krull-Schmidt Fails for Finitely Generated Modules and Artinian Modules.- 8.1 Krull-Schmidt fails for finitely generated modules.- 8.2 Krull-Schmidt fails for artinian modules.- 8.3 Notes on Chapter 8.- 9 Biuniform Modules.- 9.1 First properties of biuniform modules.- 9.2 Some technical lemmas.- 9.3 A sufficient condition.- 9.4 Weak Krull-Schmidt Theorem for biuniform modules.- 9.5 Krull-Schmidt holds for finitely presented modules over chain rings.- 9.6 Krull-Schmidt fails for finitely presented modules over serial rings.- 9.7 Further examples of biuniform modules of type 1.- 9.8 Quasi-small uniserial modules.- 9.9 A necessary condition for families of uniserial modules.- 9.10 Notes on Chapter 9.- 10 ?-pure-injective Modules and Artinian Modules.- 10.1 Rings with a faithful ?-pure-injective module.- 10.2 Rings isomorphic to endomorphism rings of artinian modules.- 10.3 Distributive modules.- 10.4 ?-pure-injective modules over chain rings.- 10.5 Homogeneous ?-pure-injective modules.- 10.6 Krull dimension and ?-pure-injective modules.- 10.7 Serial rings that are endomorphism rings of artinian modules.- 10.8 Localizable systems and ?-pure-injective modules over serial rings.- 10.9 Notes on Chapter 10.- 11 Open Problems.