## Algebra II: Noncommutative Rings IdentitiesThe algebra of square matrices of size n ~ 2 over the field of complex numbers is, evidently, the best-known example of a non-commutative alge 1 bra • Subalgebras and subrings of this algebra (for example, the ring of n x n matrices with integral entries) arise naturally in many areas of mathemat ics. Historically however, the study of matrix algebras was preceded by the discovery of quatemions which, introduced in 1843 by Hamilton, found ap plications in the classical mechanics of the past century. Later it turned out that quaternion analysis had important applications in field theory. The al gebra of quaternions has become one of the classical mathematical objects; it is used, for instance, in algebra, geometry and topology. We will briefly focus on other examples of non-commutative rings and algebras which arise naturally in mathematics and in mathematical physics. The exterior algebra (or Grassmann algebra) is widely used in differential geometry - for example, in geometric theory of integration. Clifford algebras, which include exterior algebras as a special case, have applications in rep resentation theory and in algebraic topology. The Weyl algebra (Le. algebra of differential operators with· polynomial coefficients) often appears in the representation theory of Lie algebras. In recent years modules over the Weyl algebra and sheaves of such modules became the foundation of the so-called microlocal analysis. The theory of operator algebras (Le. |

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### Contents

Contents | 4 |

2 FiniteDimensional Algebras | 29 |

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

Copyright | |

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abelian group algebra G Algebra Logic 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 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