Researches On Generalized Power Series Rings Theory

In this dissertation, we mainly study on module categories over generalized power series rings. The wholethesis is made up of four chapters.In Chapter 1, the generalized power series module is mainly discussed. Under some additional conditions,a sufficient and necessary condition for modules under which the module of generalized power series is a reducedmodule, Baer module, pp-module, quasi-Baer module, p.q.Baer module, Ikeda-Nakayama module and uniserialmodule is given, respectively. Our results unify and generalize the corresponding results of the polynomialextension and the power series extension of a module. We finish this Chapter by studying some special propertiesof rings of generalized power series.In Chapter 2, we study some special properties of generalized Macaulay-Northcott modules. We firstprove an important isomorphic formula. Using this isomorphic formula, we can not only give some new examplesof generalized power series modules, but also, by this isomorphic formula, we can prove that the pureinjective dimension of a generalized power series module less than or equal to the pure injective dimensionof its base module, and that N is a pure submodule of an R-module M if and only if [N^s,≤] is a pure submoduleof the [[R^s,≤]]- module [M^s,≤]. Secondly, the Artiness and the quasi-duality property of generalizedMacaulay-Northcott modules are discussed and, the uniform dimension and the co-uniform dimension of generalizedMacaulay-Northcott modules are considered. In the last part of this Chapter, the relationships betweengeneralized Macaulay-Northcott modules and envelopes are explored.In Chapter 3, the notion of generalized inverse power series modules is introduced. It is shown thatan R-module M is injective if and only if the generalized inverse power series module M[[S^-1]] is injective;α: E→M is an injective precover of M if and only ifα[[S^-1]]: E[[S^-1]]→M[[S^-1]] is an injective precoverof M[[S^-1]]; and that N is a pure submodule of M if and only if N[S] is a pure submodule of monoid moduleM[S].In Chapter 4, we first consider the question of triangular matrix representations of Malcev-Neumann rings.we prove that, under some additional conditions, the Malcev-Neumann ring has a complete generalized triangularmatrix representation with prime diagonal rings, and that the Malcev-Neumann ring has the same triangulatingdimension as its base ring. Secondly, in this Chapter, extensions of McCoy rings are explored. We prove thatn-by-n full matrix rings and upper triangular rings over right McCoy rings are not necessarily right McCoyrings. Simultaneously, we show that there are two special classes of n-by-n upper triangular rings over rightMcCoy rings are again right McCoy rings. We also show that the polynomial ring and the classical right quientring (if exists) over a right McCoy ring are right McCoy rings, and that for a ring A with B its subring, if A isa right McCoy ring, then R(A, B) is also a right McCoy ring.