【作者】 张俊；

【导师】 王芳贵；

【作者基本信息】 四川师范大学 ， 基础数学， 2009， 硕士

【摘要】 本文研究了交换环上w-模的升链和降链条件,引入了几个新的模类,使得一些经典的理论得到新的表现和应用.我们证明了w-Noether环上有限型模的对偶模是有限表现型的.并讨论了w-Noether模的真w-子模的准素分解问题.也研究了w-Noether环上有限型w-模的零化子,证明了w-Noether环上有限型的GV -无挠模只有有限个极大素理想,且每一个都是其中某个非零元素的零化子.其次,我们引入了w-单模的概念,不仅说明了w-单模的存在性,而且指出了w-单模与单模的差异.进而,给出了w-半单模的定义.并得到了半单环的新的刻画.证明了R是半单环当且仅当每一个w-模是w-半单模.还讨论了w-半单模的w-子模链条件的等价刻画.同时,定义了w-模的w-底座.说明了w-底座与底座是不同的两个概念.并借助w-底座得到了w-Artin模的等价条件.此外,还给出了w-模的w-Jacobson根与w-多余子模两个新概念.我们也举例说明了w-Jacobson根与一般模范畴中定义的Jacobson根相比是非平凡的,并且讨论了两者之间的关系,证明了w(R) ? J(R).还得到了关于w-Jacobson根的中山引理的相应形式.作为w-Jacobson根的应用,证明了关于w-模的Kertese定理.另外,通过定义w-加性补,进一步研究了w-模的降链条件.最后,在Krull-Remak-Schmidt定理的观点下,讨论了几类模直和分解的唯一性问题.推广了Orzech定理,得到了更一般形式的Vasconcelos定理.还讨论了w-模的Fitting引理.证明了Schur引理对于w-单模仍然成立.进而,证明了w-半单模, w-Noether环上非零的GV -无挠的内射模以及有w-合成列的w-模可以唯一分解为自同态环是局部环的不可分解子模的直和.更多还原

【Abstract】 In this paper, we investigate some properties of the ascending and descend-ing chain conditions of w-modules and introduce several new modules, so thatsome classical theories have new representation and application. Firstly, we provethat if R is a w-Noetherian ring, dual modules of finite type modules are finitelypresented type, and discuss the primary decomposition of w-submodules of w-Noetherian modules. We also study the annihilator of a non-zero element of finitetype w-modules. And it is proved that if R is a w-Noetherian ring and M is aGV -torsion-free R-module of finite type, then there exist only a finite number ofmaximal prime ideals, each of which is the annihilator of a non-zero element ofM. Secondly, we show the concept of w-simple modules. And we not only showthe existence of w-simple modules, but also point out the di?erence between w-simple modules and simple modules. Then the definition of the w-semisimplemodule is given. We also obtain new characterizations of semisimple rings andprove that R is semisimple if and only if every w-module is w-semisimple. Andwe discuss the equivalent characterizations of chain conditions of w-submodulesof w-semisimple modules. Meanwhile, we define the w-socle of a w-module, showthat the w-socle and the socle are di?erent and obtain equivalent conditions ofw-Artin modules by means of the w-socle. In addition, we show the concepts ofthe w-Jacobson radical of a w-module and the w-super?uous submodule. We alsogive an example to illustrate that in comparison to the Jacobson radical, the def-inition of the w-Jacobson radical is nontrivial and prove that w(R) ? J(R). Andwe obtain the Nakayama’s Lemma with respect to the w-Jacobson radical. Asthe application, the Kertese Theorem of w-modules is proved. By the definitionof w-addition complements, we character the descending chain conditions of w-modules. At last, we investigate the uniqueness problem of direct decompositionof several modules in view point of the Krull-Remak-Schmidt Theorem. Moreover, we generalize the Orzech Theorem and obtain a generalized Vasconcelos Theo-rem. We also discuss the Fitting Lemma of w-modules and prove that the SchurLemma is always correct for w-simple modules. Furthermore, we prove that w-semisimple modules, GV -torsion-free injective modules over a w-Noetherian ringand w-modules with w-composition series can be decomposed into a direct sumof directly indecomposable submodules with local endomorphism rings.更多还原