Classes of Modules

Presents the concepts of type dimension, a natural class, and a type submodule, by exploring how they enter into much of ring and module theory. This book develops the foundations of the subject and the advanced theorems. It is suitable for those with some background in basic ring and module theory.

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Because traditional ring theory places restrictive hypotheses on all submodules of a module, its results apply only to small classes of already well understood examples. Often, modules with infinite Goldie dimension have finite-type dimension, making them amenable to use with type dimension, but not Goldie dimension. By working with natural classes and type submodules (TS), Classes of Modules develops the foundations and tools for the next generation of ring and module theory. It shows how to achieve positive results by placing restrictive hypotheses on a small subset of the complement submodules, Furthermore, it explains the existence of various direct sum decompositions merely as special cases of type direct sum decompositions.

Carefully developing the foundations of the subject, the authors begin by providing background on the terminology and introducing the different module classes. The modules classes consist of torsion, torsion-free, s[M], natural, and prenatural. They expand the discussion by exploring advanced theorems and new classes, such as new chain conditions, TS-module theory, and the lattice of prenatural classes of right R-modules, which contains many of the previously used lattices of module classes. The book finishes with a study of the Boolean ideal lattice of a ring.

Through the novel concepts presented, Classes of Modules provides a new, unexplored direction to take in ring and module theory.