# Semidistributive Modules and Rings

Book

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

1. Front Matter
Pages i-x
Pages 1-24
Pages 25-46
Pages 47-72
Pages 73-100
Pages 101-132
Pages 133-158
Pages 159-186
Pages 187-208
Pages 209-236
Pages 237-260
Pages 261-300
Pages 301-336
14. Back Matter
Pages 337-357

### 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