Introduction to Ring Theory
Most 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.
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Linear Algebras and Artinian Rings
abelian group Artinian ring ascending chain associated automorphism basis called chain of submodules coefficients commutative ring composition series condition cyclic decomposition defined degree denoted diagonal direct product direct sum division embedded endomorphism equation equivalent Euclidean domain example Exercise expressed fc-algebra field of fractions finite group finite number finite-dimensional follows free algebra free module function functor Given hence holds homomorphism f idempotent induction injective integral domain inverse irreducible isomorphism ker f left ideal lemma linear linearly independent matrix ring maximal minimal right ideals morphism multiplication n x n nilpotent Noetherian non-zero obtain polynomial ring positive integer projective modules prove quotient R-module real numbers result right d-dependent right module right Ore domain satisfying Section semifir semisimple short exact sequence Show simple skew field skew polynomial submodule subring subset surjective tensor product Theorem unit element vector space zero