On purity in abelian groups and in modules

UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Barbara Lea Thacker (Creator)
The University of North Carolina at Greensboro (UNCG )
Web Site: http://library.uncg.edu/
Robert Bernhardt

Abstract: The purpose of this thesis is to study the elementary properties of pure subgroups of an abelian group, to use these properties to prove the Fundamental Theorem of Finitely Generated Abelian Groups, and to generalize the abelian group concept of purity to modules over a commutative ring. Many of the results of this paper are found in Kaplansky [4]. Chapter I defines a pure subgroup of an abelian group, and considers the behavior of purity with respect to direct summands, to divisible groups, to the operations of union and intersection, to homomorphisms, and to torsion and torsionfree groups. An easy method of obtaining pure subgroups of primary groups is discussed. Also, a characterization of an abelian group in which every subgroup is a pure subgroup is developed. In Chapter II, groups of bounded order are introduced. Pure subgroups which are direct sums of cyclic groups are then considered, and are used to prove that any group of bounded order is a direct sum of cyclic groups. The last major result of this chapter is using the concept of purity to prove the Fundamental Theorem of Finitely Generated Abelian Groups. Chapter III defines R-purity for modules over a commutative ring, defines injective modules, and defines the injective envelope of a module. The concept of a module being absolutely R-pure is introduced and is characterized. Several of the properties of pure subgroups of an abelian group are generalized to the module concept of R-purity.

Additional Information

Language: English
Date: 1973
Abelian groups
Rings (Algebra)

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