The Generations Of Powerseries Armendariz Modules

As a common generalization of reduced modules, Armendariz mod-ules and McCoy modules have been a hotpot in ring and module theory in recent years. This thesis will focus the study of three classes of gen-eralizations of powerseries Armendariz modules. Their properties have been studied. The thesis, consists of three chapters.In the first chapter, we mainly introduce the background of Ar-mendariz modules and McCoy modules and list the main results of this paper.In the second chapter, we firstly introduce the concept of linearly Armendariz modules and some relevant results. On the other hand, the Armendariz property of modules are used to characterize rings. It is shown that if I is a reduced ideal of a ring R and R/I is a linearly Ar-mendariz R-module, then R is linearly Armendariz. Lastly, the notion of an alpha-skew Armendariz module is introduced, and its properties are studied.In the third chapter, we introduce the concept of a powerseries Mc-Coy module, and discussed its relationship with semicommutative mod-ules and abelian modules. Direct sums (direct products) of powerseries McCoy modules are studied under some conditions. It is proved that if RR is uniform (finitely cogenerated), then direct sums(direct product- s) of a family of powerseries McCoy R-modules Mi(i ∈ F) are closed. We further consider powerseries McCoy property of polynomial exten-sions and proved the following ststements are equivalent: (1) M is a powerseries McCoy right R-module; (2) M[x] is a powerseries Mc-Coy right R[x]-module; (3) M[[x]] is a powerseries McCoy right R[[x]]-module; (4) M[x; x-1] is a powerseries McCoy right R[x; x-1]-module; (5) M[[x; x-1]] is a McCoy right R[[x; x-1]]-module.