Contents vii 6.5. Injective Modules 460 6.6. Tensor Products 469 6.7. Adjoint Isomorphisms 488 6.8. Flat Modules 493 6.9. Limits 498 6.10. Adjoint Functors 514 6.11. Galois Theory for Infinite Extensions 518 Chapter 7. Representation Theory 525 7.1. Chain Conditions 525 7.2. Jacobson Radical 534 7.3. Semisimple Rings 539 7.4. Wedderburn–Artin Theorems 550 7.5. Characters 563 7.6. Theorems of Burnside and of Frobenius 590 7.7. Division Algebras 600 7.8. Abelian Categories 614 7.9. Module Categories 626 Chapter 8. Advanced Linear Algebra 635 8.1. Modules over PIDs 635 8.1.1. Divisible Groups 646 8.2. Rational Canonical Forms 655 8.3. Jordan Canonical Forms 664 8.4. Smith Normal Forms 671 8.5. Bilinear Forms 682 8.5.1. Inner Product Spaces 682 8.5.2. Isometries 694 8.6. Graded Algebras 704 8.6.1. Tensor Algebra 706 8.6.2. Exterior Algebra 715 8.7. Determinants 729 8.8. Lie Algebras 743 Chapter 9. Homology 751 9.1. Simplicial Homology 751 9.2. Semidirect Products 757 9.3. General Extensions and Cohomology 765 9.3.1. H2(Q, K) and Extensions 766 9.3.2. H1(Q, K) and Conjugacy 774 9.4. Homology Functors 782
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