【作者】 杨国文；

【导师】 杜先能；

【作者基本信息】 安徽大学 ， 基础数学， 2010， 硕士

【摘要】 本论文主要讨论了内射模的各种推广以及它们之间的一些关系。本论文共分三章。第一章是引言,介绍了模的内射性几种推广形式的研究背景、研究历史、现状以及研究意义,其中主要包括GQP-内射模、AQP-内射模、AGP-内射模、APQ-内射模、QPQ-内射模。第二章中探讨了几种模的内射性的推广,介绍了拟GP-内射模、主拟内射模(PQ-内射模)、拟AP-内射模、几乎主拟内射模(APQ-内射模)、几乎拟主内射模(AQP-内射模)、拟主拟内射模(QPQ-内射模)、GQP-内射模的定义及其相应的等价刻画；与此同时,我们还探讨了内射模推广形式之间的相互关系,从QP-内射模及GQP-内射模的定义我们知道QP-内射模一定是GQP-内射模并且环R是右P-内射环当且仅当模RR是QP-内射模,R是GP-内射环当且仅当模RR是GQP-内射模。由于GP-内射环不一定是P-内射环,所以GQP-内射模未必是QP-内射模。第三章中探讨了两类特殊内射模GQP-内射模及AQP-内射模的一些性质,一个模的GQP-内射性以及它的自同态环的PQ-内射性、AQP-内射性以及它的AP-内射性之间的关系,从研究W(S)与J(S)的关系介绍了GQP-内射模与AQP-内射模所共有的性质,另一方面研究了S与W(S)中元素的关系以及非零M-循环子模在其中所起的串联作用。更多还原

【Abstract】 This thesis mainly investigated the properties of some classes of special injective modules and their relationships.The first chapter introduced the concepts of injective modules, such as GQP-injective modules, AQP-injective modules, AGP-injective modules, APQ-injective modules、QPQ-injective modules. And we recall some valuable results on them.In chapterⅡ, we introduced the definitions of quasi GP-injective modules、PQ-injective modules、quasi AP-injective modules、APQ-injective modules、AQP-injective modules、QPQ-injective modules、GQP-injective modules, and discussed their relations. By definitions of QP-injective modules and GQP-injective modules we know that the QP-injective modules are GQP-injective modules, the ring R is right P-injective ring if and only if the module RR is QP-injective module, and the ring R is GP-injective ring if and only if the module RR is GQP-injective module. Since GP-injective rings are not necessarily P-injective ring, GQP-injective module may not be QP-injective module.The two special injective modules, i.e. GQP-injective modules and AQP-injective modules, are dicussed in the third chapter. Meanwhile, we investigated the GQP-injectivity of a module and the PQ-injectivity of it’s endomorphism ring, the AQP-injectivity of a module and the AP-injectivity of it’s endomorphism ring. Through the study of W(S) and J(S),we know that the GQP-injective modules and AQP-injective modules shared the same property. Furthermore, we realized the M-cyclic submodule paly an important role by investigating the elements of S and W(S).更多还原