@inproceedings{10.1007/978-94-017-0337-6_10,
    Abstract = {A division ring, or skew-field, satisfies all the axioms of a field except (possibly) commutativity of multiplication. The center of a division algebra D is a field, which we always denote as F. Thus D is a vector space over F. This paper focuses on the case [D : F] < ∞; then we call D a division algebra. Division rings arise in several ways:(1)By Schur's lemma, the endomorphism ring of an arbitrary simple module is a division ring.(2)Goldie proved every noncommutative Noetherian domain is Ore, and thus has a classical ring of quotients which is a division ring. This applies for example to enveloping algebras of finite dimensional Lie algebras, and group algebras of torsion-free polycyclic-by-finite groups, rings of differential operators of nonsingular complex algebraic varieties, and other rings of current research interest in mathematics and physics.(3)Many non-Noetherian domains also can be embedded in division rings, e.g. the free ring, and enveloping algebras of arbitrary Lie algebras.(4)By the Wedderburn-Artin theorem, every simple Artinian ring is isomorphic to a matrix ring over a division ring. In particular, any finite dimensional semisimple algebra R can be written as a direct product of matrix rings {\\%} MathType!MTEF!2!1!+-{\\%} feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn{\\%} hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr{\\%} 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9{\\%} vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x{\\%} fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa{\\%} aaleaacaWG0bGaamyAaaqabaGccaGGOaGaamiramaaBaaaleaacaWG{\\%} PbaabeaakiaacMcacaGGSaaaaa!3CD9!]]</EquationSource><EquationSource Format=``TEX''><![CDATA[{\$}{\$}{\{}M{\\_}{\{}ti{\}}{\}}({\{}D{\\_}i{\}}),{\$}{\$}, for suitable division algebras Di(determined uniquely up to isomorphism). Thus division algebras arise in group representation theory, via Maschke's theorem.(5)Quadratic forms give rise to Clifford algebras, which are simple algebras, and more generally division algebras come up in the study of homogeneous forms over non-algebraically closed fields.(6)Division algebras tie in to algebraic geometry through Brauer-Severi varieties.(7)The Steinberg symbols of K-theory are intimately connected with cyclic algebras.(8)Division algebras arise in forms of algebraic groups defined over nonalgebraically closed fields.(9)Any Desarguian projective plane can be coordinatized in terms of a suitable division ring.},
    Address = {Dordrecht},
    Author = {Rowen, Louis H.},
    BookTitle = {Proceedings of the Third International Algebra Conference},
    File = {Division Aalgebras - Rowen2003\_Chapter\_DivisionAlgebras - a - y.pdf},
    ISBN = {978-94-017-0337-6},
    Pages = {221--233},
    Publisher = {Springer Netherlands},
    Title = {Division Algebras},
    Year = {2003},
    date-added = {2020-10-25 12:33:44 +0100},
    date-modified = {2020-10-25 15:17:06 +0100},
    doi = {10.1007/978-94-017-0337-6_10}
}