Research On The W-relativity Of An Arbitrary Module Pure Mathematics

In this paper, the methods of w-envelopes and w-module theory are employed to investigate an arbitrary R-module.We obtain some more general properties and results than w-module.This thesis is divided into two chapters.In Chapter 1.we firstly introduce two notions:relative w-submodule and relative w-envelope. and discuss some properties of them. Then we get that sufficient and necessary conditions for a primary submodule to be a relative w-submodule is (A:M)w,≠R or for an arbitrary GV-ideal.J∈(A:M).By the way of localization we obtain that any finite type module can be expressed as the relative w-envelope of one of its finite submodules and always has maximal relative w-submodule.At last,the main theorem of this chapter is shown that a finite type module is a w-Noether module if and only if it has the ascending chain condition on relative w-submodule.In Chapter 2.we do our work around w-superfluous submodules.We prove that simple module and GV-torsion module are GV-irreducible modules,each real relative w-submodule of w-Artin module includes a GV-irreducible submodule, and its localization about w-ideal is Artin-module. Moreover,we discussed w-superfluous submodules and w-Jacobson radicals separately. At the same time corresponding examples are also given.Finally, it is found that for the module satisfying the condition that each real relative w-submodule contained in a maximal relative w-submodule.there is a one to one correspondence between w-superfluous submodules and the submodules of w-Jacobson radical.