On structures of infinite modules

Abstract

Much of the literature on the structures of modules applies to those modules which possess a finite basis. The present paper is the development of a structure theory for particular infinite modules with countable bases. Generality of results is not as much the aim of the paper as is the application to problems concerning infinite matrices. For a commutative field P, Z is assumed to be a universal P-module which has a countable P-basis. A principal ideal ring Q which contains P is considered as an operator domain of S. Then the main topic studied is under what conditions submodules of Z have proper Q-bases. In the first place,, a complete characterization is given for the proper Q-bases of any Q-submodule of S. This is represented as an infinite matrix, and is called the characteristic matrix of the submodule. The finite case is studied in the third section. The results obtained are comparable with those of Ingraham and Wolf [3](1) and Chevalley [1]. The principal theorem is that every Q-module which possesses a finite Q-basis has a proper Q-basis. The concepts of primitivity-defined somewhat as Chevalley defines itand index play an important role in determining conditions for a Q-module to have a proper Q-basis. In order to find these conditions, the non-regular elements H of : are split from S. The resulting Q-module Z/H is regular. Then necessary and sufficient conditions are found for both H and Z/H to have a proper Q-basis. If the operator domain of Z be considered as Q/(m), m not a unit of Q, then in the fifth section it is seen that Z possesses a proper Q/(m)-basis. As an application of these results, Z is taken to be the set of all vectors over P of order type X which are finitely nonzero. The total operator domain of Z is a certain ring of infinite matrices, WV. Then any element A of YW, can be transformed into a direct sum of finite matrices only if Z has a proper P [A ]-basis. The algebraic theory assumed herein can be found in almost any book on modern algebra-specific attention is called to MacDuffee [4] and Zassenhaus [5 ].