Groups, Rings, and Modules

Abstract

This chapter presents an excursion in abstract algebra. It begins with the notions of group, subgroup, direct sum, homomorphism, isomorphism, etc., their basic properties, and numerous examples. This is justified by the main aim of the chapter: to establish the decomposition of a finite abelian group as a direct sum of cyclic subgroups, which is very similar to the decomposition of a vector space as a direct sum of cyclic subspaces (the Jordan normal form of a linear transformation). Moreover, the final part of the chapter presents a more general fact—the decomposition of a finitely generated torsion module over a Euclidean ring as a direct sum of cyclic submodules, which contains the decomposition of a vector space and the decomposition of a finite abelian group as partial cases.