Groups, Rings, Modules
This classic monograph is geared toward advanced undergraduates and graduate students. The treatment presupposes some familiarity with sets, groups, rings, and vector spaces.
The four-part approach begins with examinations of sets and maps, monoids and groups, categories, and rings. The second part explores unique factorization domains, general module theory, semisimple rings and modules, and Artinian rings. Part three's topics include localization and tensor products, principal ideal domains, and applications of fundamental theorem. The fourth and final part covers algebraic field extensions and Dedekind domains. Exercises are provided at the end of each chapter.
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abelian group arbitrary ring Basic Properties basis bilinear map Coim f commutative monoid commutative ring containing Dedekind domain deﬁnition Definition Let denote division ring endomorphism epimorphism exact sequence Example fact ﬁeld of quotients ﬁnd ﬁnite ﬁnite number ﬁnite set ﬁnite-dimensional vector space ﬁnitely generated R-module ﬁrst free R-module given Hence HomR HomR(M implies indexed family inﬁnite injective morphism integral domain invertible irreducible isomorphism of sets Ker f law of composition left ideal Lemma Let f map f map of sets matrix maximal ideal Mod(R monic polynomial monomorphism morphism f morphism of groups morphism of monoids morphism of R-modules morphism of rings multiplicative subset nonempty nonzero element phism polynomial PPD(R prime elements prime ideal principal ideal PROOF Proposition Prove R-algebra R-morphism reader ring morphism semisimple sequence of R-modules Show simple R-module submodule submonoid subring summand surjective morphism Theorem unique factorization domain unique morphism vector space