For many people there is life after 40; for some mathematicians there is algebra after Galois theory. The objective ofthis book is to prove the latter thesis. It is written primarily for students who have assimilated substantial portions of a standard first year graduate algebra textbook, and who have enjoyed the experience. The material that is presented here should not be fatal if it is swallowed by persons who are not members of that group. The objects of our attention in this book are associative algebras, mostly the ones that are finite dimensional over a field. This subject is ideal for a textbook that will lead graduate students into a specialized field of research. The major theorems on associative algebras inc1ude some of the most splendid results of the great heros of algebra: Wedderbum, Artin, Noether, Hasse, Brauer, Albert, Jacobson, and many others. The process of refine ment and c1arification has brought the proof of the gems in this subject to a level that can be appreciated by students with only modest background. The subject is almost unique in the wide range of contacts that it makes with other parts of mathematics. The study of associative algebras con tributes to and draws from such topics as group theory, commutative ring theory, field theory, algebraic number theory, algebraic geometry, homo logical algebra, and category theory. It even has some ties with parts of applied mathematics.
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ISBN-13
978-0-387-90693-5 (9780387906935)
DOI
10.1007/978-1-4757-0163-0
Schweitzer Klassifikation
1 The Associative Algebra.- 1.1. Conventions.- 1.2. Group Algebras.- 1.3. Endomorphism Algebras.- 1.4. Matrix Algebras.- 1.5. Finite Dimensional Algebras over a Field.- 1.6. Quaternion Algebras.- 1.7. Isomorphism of Quaternion Algebras.- 2 Modules.- 2.1. Change of Scalars.- 2.2. The Lattice of Submodules.- 2.3. Simple Modules.- 2.4. Semisimple Modules.- 2.5. Structure of Semisimple Modules.- 2.6. Chain Conditions.- 2.7. The Radical.- 3 The Structure of Semisimple Algebras.- 3.1. Semisimple Algebras.- 3.2. Minimal Right Ideals.- 3.3. Simple Algebras.- 3.4. Matrices of Homomorphisms.- 3.5. Wedderbum's Structure Theorem.- 3.6. Maschke's Theorem.- 4 The Radical.- 4.1. The Radical of an Algebra.- 4.2. Nakayama's Lemma.- 4.3. The Jacobson Radical.- 4.4. The Radical of an Artinian Algebra.- 4.5. Artinian Algebras Are Noetherian.- 4.6. Nilpotent Algebras.- 4.7. The Radical of a Group Algebra.- 4.8. Ideals in Artinian Algebras.- 5 Indecomposable Modules.- 5.1. Direct Decompositions.- 5.2. Local Algebras.- 5.3. Fitting's Lemma.- 5.4. The Krull-Schmidt Theorem.- 5.5. Representations of Algebras.- 5.6. Indecomposable and Irreducible Representations.- 6 Projective Modules over Artinian Algebras.- 6.1. Projective Modules.- 6.2. Homomorphisms of Projective Modules.- 6.3. Structure of Projective Modules.- 6.4. Idempotents.- 6.5. Structure of Artinian Algebras.- 6.6. Basic Algebras.- 6.7. Representation Type.- 7 Finite Representation Type.- 7.1. The Brauer-Thrall Conjectures.- 7.2. Bounded Representation Type.- 7.3. Sequence Categories.- 7.4. Simple Sequences.- 7.5. Almost Split Sequences.- 7.6. Almost Split Extensions.- 7.7. Roiter's Theorem.- 8 Representation of Quivers.- 8.1. Constructing Modules.- 8.2. Representation of Quivers.- 8.3. Application to Algebras.- 8.4. Subquivers.- 8.5. Rigid Representations.- 8.6. Change of Orientation.- 8.7. Change of Representation.- 8.8. The Quadratic Space of a Quiver.- 8.9. Roots and Representations.- 9 Tensor Products.- 9.1. Tensor Products of R-modules.- 9.2. Tensor Products of Algebras.- 9.3. Tensor Products of Modules over Algebras.- 9.4. Scalar Extensions.- 9.5. Induced Modules.- 9.6. Morita Equivalence.- 10 Separable Algebras.- 10.1. Bimodules.- 10.2. Separability.- 10.3. Separable Algebras Are Finitely Generated.- 10.4. Categorical Properties.- 10.5. The Class of Separable Algebras.- 10.6. Extensions of Separable Algebras.- 10.7. Separable Algebras over Fields.- 10.8. Separable Extensions of Algebras.- 11 The Cohomology of Algebras.- 11.1. Hochschild Cohomology.- 11.2. Properties of Cohomology.- 11.3. The Snake Lemma.- 11.4. Dimension.- 11.5. Zero Dimensional Algebras.- 11.6. The Principal Theorem.- 11.7. Split Extensions of Algebras.- 11.8. Algebras with 2-nilpotent Radicals.- 12 Simple Algebras.- 12.1. Centers of Simple Algebras.- 12.2. The Density Theorem.- 12.3. The Jacobson-Bourbaki Theorem.- 12.4. Central Simple Algebras.- 12.5. The Brauer Group.- 12.6. The Noether-Skolem Theorem.- 12.7. The Double Centralizer Theorem.- 13 Subfields of Simple Algebras.- 13.1. Maximal Subfields.- 13.2. Splitting Fields.- 13.3. Algebraic Splitting Fields.- 13.4. The Schur Index.- 13.5. Separable Splitting Fields.- 13.6. The Cartan-Brauer-Hua Theorem.- 14 Galois Cohomology.- 14.1. Crossed Products.- 14.2. Cohomology and Brauer Groups.- 14.3. The Product Theorem.- 14.4. Exponents.- 14.5. Inflation.- 14.6. Direct Limits.- 14.7. Restriction.- 15 Cyclic Division Algebras.- 15.1. Cyclic Algebras.- 15.2. Constructing Cyclic Algebras by Inflation.- 15.3. The Primary Decomposition of Cyclic Algebras.- 15.4. Characterizing Cyclic Division Algebras.- 15.5. Division Algebras of Prime Degree.- 15.6. Division Algebras of Degree Three.- 15.7. A Non-cyclic Division Algebra.- 16 Norms.- 16.1. The Characteristic Polynomial.- 16.2. Computations.- 16.3. The Reduced Norm.- 16.4. Transvections and Dilatations.- 16.5. Non-commutative Determinants.- 16.6. The Reduced Whitehead Group.- 17 Division Algebras over Local Fields.- 17.1. Valuations of Division Algebras.- 17.2. Non-archimedean Valuations.- 17.3. Valuation Rings.- 17.4. The Topology of a Valuation.- 17.5. Local Fields.- 17.6. Extension of Valuations.- 17.7. Ramification.- 17.8. Unramified Extensions.- 17.9. Norm Factor Groups.- 17.10. Brauer Groups of Local Fields.- 18 Division Algebras over Number Fields.- 18.1. Field Composita.- 18.2. More Extensions of Valuations.- 18.3. Valuations of Algebraic Number Fields.- 18.4. The Albert-Hasse-Brauer-Noether Theorem.- 18.5. The Brauer Groups of Algebraic Number Fields.- 18.6. Cyclic Algebras over Number Fields.- 18.7. The Image of INV.- 19 Division Algebras over Transcendental Fields.- 19.1. The Norm Form.- 19.2. Quasi-algebraically Closed Fields.- 19.3. Krull's Theorem.- 19.4. Tsen's Theorem.- 19.5. The Structure of B(K(x)/F(x)).- 19.6. Exponents of Division Algebras.- 19.7. Twisted Laurent Series.- 19.8. Laurent Series Fields.- 19.9. Amitsur's Example.- 20 Varieties of Algebras.- 20.1. Polynomial Identities and Varieties.- 20.2. Special Identities.- 20.3. Identities for Central Simple Algebras.- 20.4. Standard Identities.- 20.5. Generic Matrix Algebras.- 20.6. Central Polynomials.- 20.7. Structure Theorems.- 20.8. Universal Division Algebras.- References.- Index of Symbols.- Index of Terms.
