Chain Conditions On W-modules Of Commutative Rings With Zero Divisors And Application

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.