Introduction to Ring TheoryMost parts of algebra have undergone great changes and advances in recent years, perhaps none more so than ring theory. In this volume, Paul Cohn provides a clear and structured introduction to the subject. After a chapter on the definition of rings and modules there are brief accounts of Artinian rings, commutative Noetherian rings and ring constructions, such as the direct product. Tensor product and rings of fractions, followed by a description of free rings. The reader is assumed to have a basic understanding of set theory, group theory and vector spaces. Over two hundred carefully selected exercises are included, most with outline solutions. |
Contents
Basics | 9 |
Linear Algebras and Artinian Rings | 55 |
Noetherian Rings | 119 |
Ring Constructions | 151 |
General Rings | 191 |
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Common terms and phrases
a₁ abelian group Artinian ring associated automorphism b₁ basis called coefficients commutative ring composition series condition cyclic defined degree denoted diagonal direct product direct sum embedded endomorphism equation equivalent Euclidean domain example Exercise expressed field of fractions finite group finite number follows free algebra free module function functor given hence holds homomorphism f idempotent induction injective integral domain inverse irreducible isomorphism k-algebra left ideal left R-module lemma linear linearly independent M₁ mapping matrix ring maximal minimal right ideals morphism multiplication N₁ nilpotent Noetherian non-zero obtain polynomial ring projective modules Proof prove quotient R-module real numbers result right d-dependent right Ore domain right R-module satisfying Section semifir semisimple short exact sequence Show skew field skew polynomial submodule subring subset surjective tensor product Theorem u₁ unique unit element vector space Verify zero