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Some extensions of decomposition theorems in abelian groups. I. (Chinese. English summary) Zbl 1413.13009

Summary: The purpose 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.

MSC:

13C05 Structure, classification theorems for modules and ideals in commutative rings
20K35 Extensions of abelian groups
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