Algebras, Rings and Modules: Volume 1Springer Science & Business Media, 18 de jan. de 2006 - 380 páginas 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. |
Conteúdo
Decompositions of rings | 30 |
Artinian and Noetherian rings | 59 |
Categories and functors | 82 |
Projectives injectives and flats | 111 |
Homological dimensions | 143 |
Integral domains | 161 |
Dedekind domains | 189 |
Goldie rings | 210 |
Semiperfect rings | 226 |
Quivers of rings | 262 |
Serial rings and modules | 300 |
Serial rings and their properties | 319 |
Noetherian rings | 335 |
Suggestions for further reading | 365 |
Outras edições - Ver todos
Algebras, Rings and Modules, Volume 1 Michiel Hazewinkel,Nadiya Gubareni,V.V. Kirichenko Visualização parcial - 2004 |
Algebras, Rings and Modules: Non-commutative Algebras and Rings Michiel Hazewinkel,Nadiya M. Gubareni Visualização parcial - 2016 |
Algebras, Rings and Modules: Non-Commutative Algebras and Rings Michiel Hazewinkel Prévia não disponível - 2021 |
Termos e frases comuns
Abelian group 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 dimensional finite number finitely generated modules free module Hence homomorphism idempotents identity injective module integer invertible irreducible isomorphism Jacobson radical Kery lemma M₁ Math matrix maximal ideal monomorphism morphism multiplication Noetherian ring nonzero element Obviously orthogonal idempotents P₁ pairwise orthogonal Peirce decomposition permutation polynomial Pr(A prime ideal prime radical principal ideal domain projective cover projective module Proof proposition proved quiver Q quiver Q(A right A-module right ideal right Noetherian ring ring of fractions semidistributive semiperfect ring semiprime semisimple ring serial ring simple module submodule Suppose T-nilpotent theorem theory two-sided ideal two-sided Peirce decomposition uniserial vertex zero ΕΙ