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Semidistributive Modules and Rings

  • Askar A. Tuganbaev

Part of the Mathematics and Its Applications book series (MAIA, volume 449)

Table of contents

  1. Front Matter
    Pages i-x
  2. Askar A. Tuganbaev
    Pages 1-24
  3. Askar A. Tuganbaev
    Pages 25-46
  4. Askar A. Tuganbaev
    Pages 47-72
  5. Askar A. Tuganbaev
    Pages 73-100
  6. Askar A. Tuganbaev
    Pages 101-132
  7. Askar A. Tuganbaev
    Pages 133-158
  8. Askar A. Tuganbaev
    Pages 159-186
  9. Askar A. Tuganbaev
    Pages 187-208
  10. Askar A. Tuganbaev
    Pages 209-236
  11. Askar A. Tuganbaev
    Pages 237-260
  12. Askar A. Tuganbaev
    Pages 261-300
  13. Askar A. Tuganbaev
    Pages 301-336
  14. Back Matter
    Pages 337-357

About this book

Introduction

A module M is called distributive if the lattice Lat(M) of all its submodules is distributive, i.e., Fn(G + H) = FnG + FnH for all submodules F,G, and H of the module M. A module M is called uniserial if all its submodules are comparable with respect to inclusion, i.e., the lattice Lat(M) is a chain. Any direct sum of distributive (resp. uniserial) modules is called a semidistributive (resp. serial) module. The class of distributive (resp. semidistributive) modules properly cont.ains the class ofall uniserial (resp. serial) modules. In particular, all simple (resp. semisimple) modules are distributive (resp. semidistributive). All strongly regular rings (for example, all factor rings of direct products of division rings and all commutative regular rings) are distributive; all valuation rings in division rings and all commutative Dedekind rings (e.g., rings of integral algebraic numbers or commutative principal ideal rings) are distributive. A module is called a Bezout module or a locally cyclic module ifevery finitely generated submodule is cyclic. If all maximal right ideals of a ring A are ideals (e.g., if A is commutative), then all Bezout A-modules are distributive.

Keywords

DEX Division Finite Invariant Lattice Morphism Multiplication Volume algebra endomorphism ring maximum polynomial ring ring theory

Authors and affiliations

  • Askar A. Tuganbaev
    • 1
  1. 1.Moscow Power Engineering InstituteTechnological UniversityMoscowRussia

Bibliographic information