Classes of Modules: 1st Edition (Hardback) book cover

Classes of Modules

1st Edition

By John Dauns, Yiqiang Zhou

Chapman and Hall/CRC

232 pages

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pub: 2006-06-19
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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.

Reviews

"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

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

INDEX

About the Series

Chapman & Hall/CRC Pure and Applied Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT002000
MATHEMATICS / Algebra / General
MAT022000
MATHEMATICS / Number Theory