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:
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(1)
By Schur’s lemma, the endomorphism ring of an arbitrary simple module is a division ring.
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(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.
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(3)
Many non-Noetherian domains also can be embedded in division rings, e.g. the free ring, and enveloping algebras of arbitrary Lie algebras.
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(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 EquationSource % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa % aaleaacaWG0bGaamyAaaqabaGccaGGOaGaamiramaaBaaaleaacaWG % PbaabeaakiaacMcacaGGSaaaaa!3CD9! ]]</EquationSource><EquationSource Format="TEX"><
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Rowen, L.H. (2003). Division Algebras. In: Proceedings of the Third International Algebra Conference. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0337-6_10
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DOI: https://doi.org/10.1007/978-94-017-0337-6_10
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