Algebras, Rings and Modules, Volume 1
Springer Science & Business Media, Jan 18, 2006 - Mathematics - 380 pages
Accosiative rings and algebras are very interesting algebraic structures. In a strict sense, the theory of algebras (in particular, noncommutative algebras) originated fromasingleexample,namelythequaternions,createdbySirWilliamR.Hamilton in1843. Thiswasthe?rstexampleofanoncommutative”numbersystem”. During thenextfortyyearsmathematiciansintroducedotherexamplesofnoncommutative algebras, began to bring some order into them and to single out certain types of algebras for special attention. Thus, low-dimensional algebras, division algebras, and commutative algebras, were classi?ed and characterized. The ?rst complete results in the structure theory of associative algebras over the real and complex ?elds were obtained by T.Molien, E.Cartan and G.Frobenius. Modern ring theory began when J.H.Wedderburn proved his celebrated cl- si?cation theorem for ?nite dimensional semisimple algebras over arbitrary ?elds. Twenty years later, E.Artin proved a structure theorem for rings satisfying both the ascending and descending chain condition which generalized Wedderburn structure theorem. The Wedderburn-Artin theorem has since become a corn- stone of noncommutative ring theory. The purpose of this book is to introduce the subject of the structure theory of associative rings. This book is addressed to a reader who wishes to learn this topic from the beginning to research level. We have tried to write a self-contained book which is intended to be a modern textbook on the structure theory of associative rings and related structures and will be accessible for independent study.
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Decompositions of rings
Artinian and Noetherian rings
Categories and functors
Projectives injectives and flats
Quivers of rings
Serial rings and modules
Serial rings and their properties
Suggestions for further reading
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Abelian group Analogously arrows Artinian ring Boolean algebra called Consider corollary decomposed deﬁned Definition Denote descending chain diagram direct product direct sum discrete valuation ring division ring epimorphism equivalent essential extension exact sequence exists factorial ring factors ﬁeld ﬁnite finite dimensional finite number finitely generated modules free module Hence homomorphism idempotents identity injective module integer invertible irreducible isomorphism Jacobson radical left ideals Math matrix maximal ideal monomorphism morphism multiplication Nakayama’s lemma Noetherian ring nonzero element Obviously Peirce decomposition permutation polynomial Pr(A prime ideal prime radical principal ideal domain principal module projective cover projective module projective resolution Proof proposition quiver Q right A-module right Artinian right ideal right Noetherian ring ring of fractions semidistributive semiperfect ring semiprime semisimple ring serial ring simple module submodule sum of pairwise Suppose T-nilpotent theory two-sided ideal two-sided Peirce decomposition uniserial vertex zero