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Hungerford - Algebra


Springer GTM 73.

Algebra

"This book is intended to serve as a basic text for an algebra course at the beginning graduate level. Its writing was begun several years ago when I was unable to find a one-volume text which I considered suitable for such a course."

Introduction Edit

Chapter I: Groups Edit

Chapter II: The Structure of Groups Edit

Chapter III: Rings Edit

Chapter IV: Modules Edit

Section IV.1: Modules, Homomorphisms, and Exact Sequences Edit

Section IV.2: Free Modules and Vector Spaces Edit

Section IV.3: Projective and Injective Modules Edit

Section IV.4: Hom and Duality Edit

Section IV.5: Tensor Products Edit

Section IV.6: Modules over a Principal Ideal Domain Edit

  • Proof of Theorem 6.1: If $ c \neq 0 $, then the R-module epimorphism $ R \mapsto Rc $ of Theorem 1.5(i) is actually an isomorphism. Since R is an integral domain, the kernel of this map is zero, so the map is injective. Consequently, any ideal I of a PID R is isomorphic, as an R-module, to R. (The potentially frightening implications in the finite case are dismissed by recalling that finite integral domains are already fields.)

Section IV.7: Algebras Edit

Chapter V: Fields and Galois Theory Edit

Chapter VI: The Structure of Fields Edit

Chapter VII: Linear Algebra Edit

Chapter VIII: Commutative Rings and Modules Edit

Chapter IX: The Structure of Rings Edit

Chapter X: Categories Edit