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.
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.Hinweis: Dieser Artikel kann nur an eine deutsche Lieferadresse ausgeliefert werden.
Askar Tuganbaev received his Ph.D. at the Moscow State University in 1978 and has been a professor at Moscow Power Engineering Institute (Technological University) since 1978. He is the author of three other monographs on ring theory and has written numerous articles on ring theory.
Inhaltsangabe
Introduction. Symbols. 1. Radicals, Local and Semisimple Modules. 2. Projective and Injective Modules. 3. Bezout and Regular Modules. 4. Continuous and Finite-Dimensional Modules. 5. Rings of Quotients. 6. Flat Modules and Semiperfect Rings. 7. Semihereditary and Invariant Rings. 8. Endomorphism Rings. 9. Distributive Rings with Maximum Conditions. 10. Self-Injective and Skew-Injective Rings. 11. Semidistributive and Serial Rings. 12. Monoid Rings and Related Topics. Bibliography. Index.
1 Radicals, local and semisimple modules.- 1.1 Maximal submodules and the Jacobson radical.- 1.2 Local and uniserial modules.- 1.3 Semisimple and Artinian modules.- 1.4 The prime radical.- 2 Projective and injective modules.- 2.1 Free and projective modules.- 2.2 Injective modules.- 2.3 Injective hull.- 3 Bezout and regular modules.- 3.1 Regular modules.- 3.2 Unit-regular rings.- 3.3 Semilocal rings and distributivity.- 3.4 Strongly regular rings.- 3.5 Bezout rings.- 4 Continuous and finite-dimensional modules.- 4.1 Closed submodules.- 4.2 Continuous modules.- 4.3 Finile-dimensional modules.- 4.4 Nonsingular ?-injective modules.- 5 Rings of quotients.- 5.1 Ore sets.- 5.2 Denominator sets and localizable rings.- 5.3 Maximal rings of quotients.- 6 Flat modules and semiperfect rings.- 6.1 Characterizations of flat modules.- 6.2 Submodules of flat modules.- 6.3 Semiperfect and perfect rings.- 7 Semihereditary and invariant rings.- 7.1 Coherent and reduced rings.- 7.2 Invariant rings.- 7.3 Rings with integrally closed factor rings.- 8 Endomorphism rings.- 8.1 Modules over endomorphism rings and quasi injective modules.- 8.2 Nilpotent endomorphisms.- 8.3 Strongly indecomposable modules.- 9 Distributive rings with maximum conditions.- 9.1 Arithmetics of ideals.- 9.2 Noel.herian rings.- 9.3 Classical rings of quotients of distributive rings.- 9.4 Rings algebraic over their centre.- 10 Self-injective and skew-injective rings.- 10.1 Quasi-frobenius rings and direct sums of injective modules.- 10.2 Cyclic ?-injective modules.- 10.3 Integrally closed Noetherian rings.- 10.4 Cyclic skew-injective modules.- 10.5 Countably injective rings.- 11 Semidistributive and serial rings.- 11.1 Semidistributive modules.- 11.2 Semidistributive rings.- 11.3 Serial modules and rings.- 12 Monoidrings and related topics.- 12.1 Series and polynomial rings.- 12.2 Quaternion algebras.- 12.3 Subgroups, submonoids, and annihilators.- 12.4 Regular group rings.- 12.5 Cancellative monoids.- 12.6 Semilattices and regular monoids.
Introduction. Symbols. 1. Radicals, Local and Semisimple Modules. 2. Projective and Injective Modules. 3. Bezout and Regular Modules. 4. Continuous and Finite-Dimensional Modules. 5. Rings of Quotients. 6. Flat Modules and Semiperfect Rings. 7. Semihereditary and Invariant Rings. 8. Endomorphism Rings. 9. Distributive Rings with Maximum Conditions. 10. Self-Injective and Skew-Injective Rings. 11. Semidistributive and Serial Rings. 12. Monoid Rings and Related Topics. Bibliography. Index.
1 Radicals, local and semisimple modules.- 1.1 Maximal submodules and the Jacobson radical.- 1.2 Local and uniserial modules.- 1.3 Semisimple and Artinian modules.- 1.4 The prime radical.- 2 Projective and injective modules.- 2.1 Free and projective modules.- 2.2 Injective modules.- 2.3 Injective hull.- 3 Bezout and regular modules.- 3.1 Regular modules.- 3.2 Unit-regular rings.- 3.3 Semilocal rings and distributivity.- 3.4 Strongly regular rings.- 3.5 Bezout rings.- 4 Continuous and finite-dimensional modules.- 4.1 Closed submodules.- 4.2 Continuous modules.- 4.3 Finile-dimensional modules.- 4.4 Nonsingular ?-injective modules.- 5 Rings of quotients.- 5.1 Ore sets.- 5.2 Denominator sets and localizable rings.- 5.3 Maximal rings of quotients.- 6 Flat modules and semiperfect rings.- 6.1 Characterizations of flat modules.- 6.2 Submodules of flat modules.- 6.3 Semiperfect and perfect rings.- 7 Semihereditary and invariant rings.- 7.1 Coherent and reduced rings.- 7.2 Invariant rings.- 7.3 Rings with integrally closed factor rings.- 8 Endomorphism rings.- 8.1 Modules over endomorphism rings and quasi injective modules.- 8.2 Nilpotent endomorphisms.- 8.3 Strongly indecomposable modules.- 9 Distributive rings with maximum conditions.- 9.1 Arithmetics of ideals.- 9.2 Noel.herian rings.- 9.3 Classical rings of quotients of distributive rings.- 9.4 Rings algebraic over their centre.- 10 Self-injective and skew-injective rings.- 10.1 Quasi-frobenius rings and direct sums of injective modules.- 10.2 Cyclic ?-injective modules.- 10.3 Integrally closed Noetherian rings.- 10.4 Cyclic skew-injective modules.- 10.5 Countably injective rings.- 11 Semidistributive and serial rings.- 11.1 Semidistributive modules.- 11.2 Semidistributive rings.- 11.3 Serial modules and rings.- 12 Monoidrings and related topics.- 12.1 Series and polynomial rings.- 12.2 Quaternion algebras.- 12.3 Subgroups, submonoids, and annihilators.- 12.4 Regular group rings.- 12.5 Cancellative monoids.- 12.6 Semilattices and regular monoids.
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