【作者】 胡江胜；

【导师】 陈淼森；

【作者基本信息】 浙江师范大学 ， 基础数学， 2010， 硕士

【摘要】 本学位论文共分为三章：第一章引入了PI-内射模与PI-投射模,证明了PI-投射模类与PI-内射模类构成一个完备的余挠对,借助这些我们定义了弱完全环并给出了Noether环、von Neumann正则环和半单环的一些新刻画.第二章首先证明了第一章构造的余挠对是遗传的当且仅当环R的每个纯右理想是PI-投射的.进一步地,在每个纯右理想都是PI-投射模的环R上,定义了PI-投射维数与PI-内射维数.我们发现PI-投射维数与PI-内射维数能分别度量一个环与von Neumann正则环与弱完全环之间的差距.第三章我们证明了在交换Artin环上,每个R-模都有强Copure-内射复盖.同时我们利用Hom的左导出函子及R-模的左强Copure-内射分解刻画了模与环的Copure-内射维数.更多还原

【Abstract】 This thesis is made up of three chapters:In Chapter 1, the notions of PI-injective module and PI-projective module are introduced. With the help of these objects, we prove that the class of PI-injective mod-ules and the class of PI-projective modules can be constructed to be a completely cotor-sion pair. Furthermore, we define the weakly perfect rings and give some new charac-terizations for some well-known rings, such as von Neumann regular rings, noetherian rings and semisimple Artinian rings.In Chapter 2, the cotorsion pair constructed in Chapter 1 is hereditary if and only if every pure right ideal I of R is PI-projective is proved at first. Furthermore, PI-projective dimension and PI-injective dimension for modules and rings are defined respectively and it turns out that PI-projective dimension and PI-injective dimension measure how far away a ring is from being von Neumann regular and weakly perfect respectively provided that every pure right ideal of R is PI-projective.In Chapter 3, the existence of the strongly copure injective cover is proved over a commutative artinian ring at first. Furthermore, we discuss the copure injective dimen-sions of modules and rings in terms of the left derived functor of Hom.更多还原