Ebook: 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.

---

---

Content:
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

---

Content:
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
....