Preface to Second Edition xi introduce Brauer groups. Chapter 9 views Brauer groups as cohomology groups, allowing us to prove existence of division rings of positive characteristic. This chap- ter ends with abelian categories and the characterization of categories of modules, enabling us to see why a ring R is intimately related to the matrix rings Matn(R) for n ≥ 1. Chapter 8 considers finitely generated modules over principal ideal domains (generalizing earlier theorems about finite abelian groups), and then goes on to apply these results to rational, Jordan, and Smith canonical forms for matrices over a field (the Smith normal form yields algorithms that compute elementary divisors of matrices). We also classify projective, injective, and flat modules over PIDs. Bilinear forms are introduced, along with orthogonal and symplectic groups. We then discuss some multilinear algebra, ending with an introduction to Lie Algebra. Chapter 9 introduces homological methods, beginning with abstract simpli- cial complexes and group extensions, which motivate homology and cohomology groups, and Tor and Ext (existence of these functors is proved with derived func- tors). Applications are made to modules, cohomology of groups, and division rings. A descriptive account of spectral sequences is given, suﬃcient to indicate why Ext and Tor are independent of the variable resolved. We then pass from Homolog- ical Algebra to Homotopical Algebra: Algebraic K-Theory is introduced with a discussion of Grothendieck groups. Chapter 10 returns to Commutative Algebra: localization, the general Null- stellensatz (using Jacobson rings), Dedekind rings and some Algebraic Number Theory, and Krull’s Principal Ideal Theorem. We end with more Homological Algebra, proving the Serre–Auslander–Buchsbaum Theorem characterizing regu- lar local rings as those noetherian local rings of finite global dimension and the Auslander–Buchsbaum Theorem that regular local rings are UFDs. Each generation should survey Algebra to make it serve the present time. It is a pleasure to thank the following mathematicians whose suggestions have greatly improved my original manuscript: Robin Chapman, Daniel R. Grayson, Ilya Kapovich, T.-Y. Lam, David Leep, Nick Loehr, Randy McCarthy, Patrick Szuta, and Stephen Ullom and I give special thanks to Vincenzo Acciaro for his many comments, both mathematical and pedagogical, which are incorporated throughout the text. Joseph Rotman Urbana, IL, 2009

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