Classes of Modules

Classes of Modules

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Description

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.show more

Product details

  • Hardback | 232 pages
  • 157.5 x 243.8 x 17.8mm | 430.92g
  • Taylor & Francis Ltd
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 1584886609
  • 9781584886600

Table of contents

PRELIMINARY BACKGROUND Notation and Terminology Lattices IMPORTANT MODULE CLASSES AND CONSTRUCTIONS Torsion Theory The Module Class s[M] Natural Classes M-Natural Classes Pre-Natural Classes FINITENESS CONDITIONS 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 TS-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 REFERENCES INDEXshow more