Abel群的一些分解定理的推广(Ⅰ)

Some Extensions of Decomposition Theorems In Abelian Groups(Ⅰ)

这项研究的目的是要把Abel群(有限或无限)的诸多分解定理尽可能地推广到主理想整环的模上,得到这类模上的分解定理,随后再把所得定理应用到向量空间(有限维或无限维)及其线性变换,得到向量空间的分解定理.本文是系列文章的第一篇,主要目的是建立起支撑整个研究的最基本概念,例如纯子模、有界模、局部循环模、具有minimax条件的模等.本文主要内容有:(1)确定了主理想整环上可除模、有界模、局部循环模的结构;(2)给出了主理想整环上拟循环模的生成性质,这类模在以后的研究里起着非常重要的作用;(3)描述了主理想整环上满足极小条件,minimax条件的模的结构;(4)给出了两个不同构的Z[i]-模,它们作为Abel群是同构的.

The purposes of this study is to extend as many decomposition theorems of abelian groups （finite or infinite） as possible to modules over a principal ideal domain （PID） and then using these theorems of modules to study vector spaces and linear transformations on vector spaces and hence obtain decomposition theorems of vector spaces （finite or infinite dimension）. As the first of a series of articles, this paper aims to present basic concepts for the whole study, such as pure submodule, bounded module, locally cyclic module etc. The main content of this paper is as follows： （1） determining the structures of divisible modules, bounded modules and locally cyclic modules over a PID; （2） giving generating properties of the quasicyclic module over a PID, which plays a very important role in the future study; （3） characterizing modules over a PID with minimal condition or with minimax condition; （4） offering two nonisomorphic Z[i]-modules, but isomorphic as Abelian groups.