Author | Pierce, Richard S. author |
---|---|
Title | Associative Algebras [electronic resource] / by Richard S. Pierce |
Imprint | New York, NY : Springer New York, 1982 |
Connect to | http://dx.doi.org/10.1007/978-1-4757-0163-0 |
Descript | 436p. online resource |
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. Wedderbumโs Structure Theorem -- 3.6. Maschkeโs Theorem -- 4 The Radical -- 4.1. The Radical of an Algebra -- 4.2. Nakayamaโs 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. Fittingโs 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. Roiterโs 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. Krullโs Theorem -- 19.4. Tsenโs 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. Amitsurโs 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