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Introduction to Division Rings [PDF]
, 1991 The study of division rings at the beginning stage is a wonderland of beautiful results. It all started with Frobenius’ Theorem (c. 1877) classifying the finite-dimensional division algebras over the reals. Nowadays, we know that the theorem also works for algebraic algebras. Shortly after E. H. Moore completed his classification of finite fields, J. H.
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A Survey on Free Objects in Division Rings and in Division Rings with an Involution
Communications in Algebra, 2012Let D be a division ring with center k, and let D † be its multiplicative group. We investigate the existence of free groups in D †, and free algebras and free group algebras in D. We also go through the case when D has an involution * and consider the existence of free symmetric and unitary pairs in D †.
M. Shirvani, Jairo Z. Gonçalves
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Group Rings and Division Rings [PDF]
, 1984 Continuing the work in [ll],[l2] we study division algebras D = k(G) over a field k which are generated by some polycyclic-by-finite subgroup G of the multiplicative group D* of D. We discuss a specific class of examples of such division algebras that can be thought of as multiplicative analogs of the Weyl field.
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Canadian Journal of Mathematics, 1951
The object of this note is to prove the following theorem. THEOREM. Let A be a division ring with centre Z, and suppose that for every x in A, some power (depending on x) is in . Then A is commutative.
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The object of this note is to prove the following theorem. THEOREM. Let A be a division ring with centre Z, and suppose that for every x in A, some power (depending on x) is in . Then A is commutative.
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Alternative Division Rings, II [PDF]
, 2002 In this chapter we prove Theorem 17.3. Our goal is to show that the Cayley-Dickson algebras defined in (9.8) are the only non-associative alternative division rings. This result was first proved in [17] and [56] by R. Bruck and E. Kleinfeld. See also [3], [74] and [87]. In the proof we give here, the characteristic does not play any role.
Richard M. Weiss, Jacques Tits
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Theories of rings with divisibility
Communications in Algebra, 1997We consider several different theories of commutative rings with a new predicate symbol | together with the additional axiom a|b ⇔ ∃c (a . c = b) and show that none of these has a model companion.
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Israel Journal of Mathematics, 1974
LetA=k (X 1, X2..., Xm) be the division ring generated by genericn×n matrices over a fieldk; thenA is not a crossed product in the following cases: (i) there exists a primeq such thatq 3ℛn;(ii)[k:Q]=m, whereQ is the field of rationals, then if eitherq 3ℛn for someq for whichq-1ℛm, orq 2/nn
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LetA=k (X 1, X2..., Xm) be the division ring generated by genericn×n matrices over a fieldk; thenA is not a crossed product in the following cases: (i) there exists a primeq such thatq 3ℛn;(ii)[k:Q]=m, whereQ is the field of rationals, then if eitherq 3ℛn for someq for whichq-1ℛm, orq 2/nn
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Israel Journal of Mathematics, 1983
Letk be a field. WriteD(G) for the quotient division ring of the group ringkG of a torsion-free, polycyclic-by-finite groupG, andD(g) for the quotient ring of the enveloping algebra of a finite-dimensional Lie algebrag overk. In this note we show that the Hirsch numberh(G) and dim k g are invariants for
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Letk be a field. WriteD(G) for the quotient division ring of the group ringkG of a torsion-free, polycyclic-by-finite groupG, andD(g) for the quotient ring of the enveloping algebra of a finite-dimensional Lie algebrag overk. In this note we show that the Hirsch numberh(G) and dim k g are invariants for
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