Semidistributive Modules and Rings

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. 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 Front Matter....Pages i-x Radicals, local and semisimple modules....Pages 1-24 Projective and injective modules....Pages 25-46 Bezout and regular modules....Pages 47-72 Continuous and finite-dimensional modules....Pages 73-100 Rings of quotients....Pages 101-132 Flat modules and semiperfect rings....Pages 133-158 Semihereditary and invariant rings....Pages 159-186 Endomorphism rings....Pages 187-208 Distributive rings with maximum conditions....Pages 209-236 Self-injective and skew-injective rings....Pages 237-260 Semidistributive and serial rings....Pages 261-300 Monoid rings and related topics....Pages 301-336 Back Matter....Pages 337-357 This is the first monograph on the theory of semidistributive modules and rings. It investigates such topics as the relationship between semidistributive modules and flat, projective, injective, multiplication, as well as Bezout modules. The volume concludes with an extensive bibliography. Audience: This work can be recommended as an introduction to structural and homological ring theory, and will prove useful for postgraduates and researchers specialising in algebra.