1. Structure of Rings.- 1.1 Baer Radical and Semiprimeness.- 1.2 Automorphism Groups and Lie Differential Algebras.- 1.3 Bergman-Isaacs Theorem. Shelter Integrality.- 1.4 Martindale Ring of Quotients.- 1.5 The Generalized Centroid of a Semiprime Ring.- 1.6 Modules over a Generalized Centroid.- 1.7 Extension of Automorphisms to a Ring of Quotients. Conjugation Modules.- 1.8 Extension of Derivations to a Ring of Quotients.- 1.9 The Canonical Sheaf of a Semiprime Ring.- 1.10 Invariant Sheaves.- 1.11 The Metatheorem.- 1.12 Stalks of Canonical and Invariant Sheaves.- 1.13 Martindale's Theorem.- 1.14 Quite Primitive Rings.- 1.15 Rings of Quotients of Quite Primitive Rings.- 2. On Algebraic Independence of Automorphisms And Derivations.- 2.0 Trivial Algebraic Dependences.- 2.1 The Process of Reducing Polynomials.- 2.2 Linear Differential Identities with Automorphisms.- 2.3 Multilinear Differential Identities with Automorphisms.- 2.4 Differential Identities of Prime Rings.- 2.5 Differential Identities of Semiprime Rings.- 2.6 Essential Identities.- 2.7 Some Applications: Galois Extentions of Pi-Rings; Algebraic Automorphisms and Derivations; Associative Envelopes of Lie-Algebras of Derivations.- 3. The Galois Theory of Prime Rings (The Case Of Automorphisms).- 3.1 Basic Notions.- 3.2 Some Properties of Finite Groups of Outer Automorphisms.- 3.3 Centralizers of Finite-Dimensional Algebras.- 3.4 Trace Forms.- 3.5 Galois Groups.- 3.6 Maschke Groups. Prime Dimensions.- 3.7 Bimodule Properties of Fixed Rings.- 3.8 Ring of Quotients of a Fixed Ring.- 3.9 Galois Subrings for M-Groups.- 3.10 Correspondence Theorems.- 3.11 Extension of Isomorphisms.- 4. The Galois Theory of Prime Rings (The Case Of Derivations).- 4.1 Duality for Derivations in the Multiplication Algebra.- 4.2Transformation of Differential Forms.- 4.3 Universal Constants.- 4.4 Shirshov Finiteness.- 4.5 The Correspondence Theorem.- 4.6 Extension of Derivations.- 5. The Galois Theory of Semiprime Rings.- 5.1 Essential Trace Forms.- 5.2 Intermediate Subrings.- 5.3 The Correspondence Theorem for Derivations.- 5.4 Basic Notions of the Galois Theory of Semiprime Rings (the case of automorphisms).- 5.5 Stalks of an Invariant Sheaf for a Regular Group. Homogenous Idempotents.- 5.6 Principal Trace Forms.- 5.7 Galois Groups.- 5.8 Galois Subrings for Regular Closed Groups.- 5.9 Correspondence and Extension Theorems.- 5.10 Shirshov Finiteness. The Structure of Bimodules.- 6. Applications.- 6.1 Free Algebras.- 6.2 Noncommutative Invariants.- 6.3 Relations of a Ring with Fixed Rings.- 6.4 Relations of a Semiprime Ring with Ring of Constants.- 6.5 Hopf Algebras.- References.
Inhaltsverzeichnis
1. Structure of Rings. - 1. 1 Baer Radical and Semiprimeness. - 1. 2 Automorphism Groups and Lie Differential Algebras. - 1. 3 Bergman-Isaacs Theorem. Shelter Integrality. - 1. 4 Martindale Ring of Quotients. - 1. 5 The Generalized Centroid of a Semiprime Ring. - 1. 6 Modules over a Generalized Centroid. - 1. 7 Extension of Automorphisms to a Ring of Quotients. Conjugation Modules. - 1. 8 Extension of Derivations to a Ring of Quotients. - 1. 9 The Canonical Sheaf of a Semiprime Ring. - 1. 10 Invariant Sheaves. - 1. 11 The Metatheorem. - 1. 12 Stalks of Canonical and Invariant Sheaves. - 1. 13 Martindale s Theorem. - 1. 14 Quite Primitive Rings. - 1. 15 Rings of Quotients of Quite Primitive Rings. - 2. On Algebraic Independence of Automorphisms And Derivations. - 2. 0 Trivial Algebraic Dependences. - 2. 1 The Process of Reducing Polynomials. - 2. 2 Linear Differential Identities with Automorphisms. - 2. 3 Multilinear Differential Identities with Automorphisms. - 2. 4 Differential Identities of Prime Rings. - 2. 5 Differential Identities of Semiprime Rings. - 2. 6 Essential Identities. - 2. 7 Some Applications: Galois Extentions of Pi-Rings; Algebraic Automorphisms and Derivations; Associative Envelopes of Lie-Algebras of Derivations. - 3. The Galois Theory of Prime Rings (The Case Of Automorphisms). - 3. 1 Basic Notions. - 3. 2 Some Properties of Finite Groups of Outer Automorphisms. - 3. 3 Centralizers of Finite-Dimensional Algebras. - 3. 4 Trace Forms. - 3. 5 Galois Groups. - 3. 6 Maschke Groups. Prime Dimensions. - 3. 7 Bimodule Properties of Fixed Rings. - 3. 8 Ring of Quotients of a Fixed Ring. - 3. 9 Galois Subrings for M-Groups. - 3. 10 Correspondence Theorems. - 3. 11 Extension of Isomorphisms. - 4. The Galois Theory of Prime Rings (The Case Of Derivations). - 4. 1 Duality for Derivations in the Multiplication Algebra. - 4. 2Transformation of Differential Forms. - 4. 3 Universal Constants. - 4. 4 Shirshov Finiteness. - 4. 5 The Correspondence Theorem. - 4. 6 Extension of Derivations. - 5. The Galois Theory of Semiprime Rings. - 5. 1 Essential Trace Forms. - 5. 2 Intermediate Subrings. - 5. 3 The Correspondence Theorem for Derivations. - 5. 4 Basic Notions of the Galois Theory of Semiprime Rings (the case of automorphisms). - 5. 5 Stalks of an Invariant Sheaf for a Regular Group. Homogenous Idempotents. - 5. 6 Principal Trace Forms. - 5. 7 Galois Groups. - 5. 8 Galois Subrings for Regular Closed Groups. - 5. 9 Correspondence and Extension Theorems. - 5. 10 Shirshov Finiteness. The Structure of Bimodules. - 6. Applications. - 6. 1 Free Algebras. - 6. 2 Noncommutative Invariants. - 6. 3 Relations of a Ring with Fixed Rings. - 6. 4 Relations of a Semiprime Ring with Ring of Constants. - 6. 5 Hopf Algebras. - References.
