This book presents modern algebra from first principles and is accessible to undergraduates or graduates. It combines standard materials and necessary algebraic manipulations with general concepts that clarify meaning and importance. This conceptual approach to algebra starts with a description of algebraic structures by means of axioms chosen to suit the examples, for instance, axioms for groups, rings, fields, lattices, and vector spaces. This axiomatic approach - emphasized by Hilbert and developed in Germany by Noether, Artin, Van der Waerden, et al., in the 1920s - was popularized for the graduate level in the 1940s and 1950s to some degree by the authors' publication of A Survey of Modern Algebra. The present book presents the developments from that time to the first printing of this book. This third edition includes corrections made by the authors.
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abelian group addition arrow assignment automorphism axioms basis bijection binary operation biproduct called coefficients column commutative ring complex numbers composite conjugate construct contravariant coordinates Corollary coset defined definition diagonal diagram divisor dual E X E R C I S E S elementary endomorphism epimorphism equation equivalent example factors field F formula free module function f functor given graded algebra graded module group G hence identity implies induction inner product space integers inverse irreducible isomorphism kernel lattice LEMMA linear combination linear form linear map linear transformation linearly independent matrix minimal polynomial monoid monomorphism morphism n x n natural numbers non-zero normal subgroup orthogonal orthonormal pair permutation polynomial f poset prime principal ideal Proof Proposition prove quadratic form quotient group R-module real numbers scalar multiple sequence spanned splitting field submodule subset subspace symmetric tensor product Theorem universal element vector space zero