~t moi, ... si favait su comment en revenir. One sel'Yice mathematics has rendered the je n'y serais point aile.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense', able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a…mehr
~t moi, ... si favait su comment en revenir. One sel'Yice mathematics has rendered the je n'y serais point aile.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense', able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d 'ere of this series.
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.
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.
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