## Algebra II: noncommutative rings, identities, Volume 2This book is wholeheartedly recommended to every student or user of mathematics. Although the author modestly describes his book as 'merely an attempt to talk about' algebra, he succeeds in writing an extremely original and highly informative essay on algebra and its place in modern mathematics and science. From the fields, commutative rings and groups studied in every university math course, through Lie groups and algebras to cohomology and category theory, the author shows how the origins of each algebraic concept can be related to attempts to model phenomena in physics or in other branches of mathematics. Comparable in style with Hermann Weyl's evergreen essay The Classical Groups, Shafarevich's new book is sure to become required reading for mathematicians, from beginners to experts. |

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### Contents

Contents | 4 |

2 FiniteDimensional Algebras | 29 |

Kostrikin I R Shafarevich Steklov Mathematical Institute ul Vavilova | 42 |

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abelian group algebra G Algebra Logika Artinian ring associative algebras associative rings automorphisms Bakhturin base ﬁeld basis of identities Brauer group called central simple algebra characteristic zero classical commutative ring contains countable deﬁning relations deﬁnition denote diagram direct sum division ring elements English translation equivalent example exists ﬁeld ﬁeld F ﬁeld of characteristic ﬁnd ﬁnite ﬁnite basis ﬁnite group ﬁnite-dimensional algebra ﬁrst ﬁxed free algebra free group group algebra group G homomorphism indecomposable inﬁnite injective integer irreducible isomorphic Jacobson radical Krull dimension lattice left ideal left R-module Lie algebras linear algebras Mal’tsev mapping Math matrices module multiplication Nauk Neumann nilpotent Noetherian ring non-commutative non-trivial non-zero obtained operation polynomial prime rings problem R-module Razmyslov regular ring representations result satisﬁes Sect semigroup semiprime semisimple solvable subalgebra submodules subset subvarieties system of identities topological varieties of groups variety of Lie vector space zero divisors