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.
Table of Contents
Notation and Terminology
IMPORTANT MODULE CLASSES AND CONSTRUCTIONS
The Module Class s[M]
Ascending Chain Conditions
Descending Chain Conditions
Covers and Ascending Chain Conditions
TYPE THEORY OF MODULES: DIMENSION
Type Submodules and Type Dimensions
Several Type Dimension Formulas
Some Non-Classical Finiteness Conditions
TYPE THEORY OF MODULES: DECOMPOSITIONS
Type Direct Sum Decompositions
Decomposability of Modules
Unique Type Closure Modules
LATTICES OF MODULE CLASSES
The Lattice of Pre-Natural Classes
More Sublattice Structures
Lattice Properties of Npr (R)
More Lattice Properties of Npr (R)
The Lattice Nr(R) and Its Applications
The Boolean Ideal Lattice
"This nice book is written in a very clear and explanatory style, offering a self-contained presentation as well as illustrative examples, and demonstrating how the themes of (pre-) natural classes and type submodules structure much of Ring and Module Theory. I believe that I should be on the desk of anybody working in this area of Algebra."
– Toma Albu, in Mathematical Reviews, 2007m