Free Products of Units in Algebras I. Quaternion Algebras

Gonçalves, Jairo Z.; Mandel, Arnaldo; Shirvani, M.

doi:10.1006/jabr.1998.7680

Cited by 21 publications

(19 citation statements)

References 17 publications

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“…Then H p is the four-dimensional F -algebra with basis 1, i, j, k subject to the relations i 2 = a, j 2 = b, ij = −ji = k. We will sometimes denote this algebra by the symbol ( Proof. We know that 1 ±i and 1 ±j are units in H p and that 1 +i, 1 +j is free of rank 2 by [5,Proposition 16]. Thus H = (1 +i) 2 , (1 +j) 2 is also free of rank 2.…”

Section: Crossed Product Resultsmentioning

confidence: 90%

Free groups in normal subgroups of the multiplicative group of a division ring

Gonçalves

Passman

2015

Journal of Algebra

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Let D be a division ring with center Z and multiplicative group D \ {0} = D • , and let N be a normal subgroup of D • . We investigate various conditions under which N must contain a free noncyclic subgroup. In one instance, assuming that the transcendence degree of Z over its prime field is infinite, and that N contains a nonabelian solvable subgroup, we use a construction method due to Chiba to exhibit free generators of the free subgroup.

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Section: Crossed Product Resultsmentioning

confidence: 90%

Free groups in normal subgroups of the multiplicative group of a division ring

Gonçalves

Passman

2015

Journal of Algebra

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“…In [4,8,9,11], and some other papers, for various rings R explicit copies of F ⊂ U(R) were found. We consider the same problem for orders, but not only Z-orders, in finite-dimensional, semisimple Q-algebras.…”

Section: Motivation and Main Resultmentioning

confidence: 99%

“…Lately, in [4], a more general, but more complicated approach to effective construction of free subgroups of units in quaternion algebras was found. This approach is different from that presented here and, for example, has the result ofŚwierczkowski from [15] only as a consequence of the main theorems.…”

Section: Sketch Of Proofmentioning

confidence: 99%

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On free subgroups of units in quaternion algebras

Krempa

2001

Colloq. Math.

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Abstract. It is well known that for the ring H(Z) of integral quaternions the unit group U(H(Z)) is finite. On the other hand, for the rational quaternion algebra H(Q), its unit group is infinite and even contains a nontrivial free subgroup. In this note (see Theorem 1.5 and Corollary 2.6) we find all intermediate rings Z ⊂ A ⊆ Q such that the group of units U(H(A)) of quaternions over A contains a nontrivial free subgroup. In each case we indicate such a subgroup explicitly. We do our best to keep the arguments as simple as possible. Motivation and main result.In this paper N ⊂ Z ⊂ Q ⊂ R have standard meaning as subsets of the field C of complex numbers, U(R) denotes the group of units of any associative ring R with 1 = 0, F stands for a free nonabelian group with two generators, and SO n (R) ⊂ GL n (R) for the well known linear groups over a given ring R. We also apply some other standard notation and terminology (see for example [10,13,17]).In [4,8,9,11], and some other papers, for various rings R explicit copies of F ⊂ U(R) were found. We consider the same problem for orders, but not only Z-orders, in finite-dimensional, semisimple Q-algebras. An old and simple, but very useful and effective result in this direction, due to Sanov and Brenner (see [17]), is:As a consequence of this result we have Theorem 1.2. Let R be any order in a finite-dimensional semisimple Q-algebra. If R has a nonzero nilpotent element then there exists an effective way to construct a copy of F ⊆ U(R).

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“…This question has been an object of renewed interest in recent years. For example, in Gonçalves et al (1999); Gonçalves et al (2000) Mandel and the present authors looked at the question for crossed product algebras. The first author, together with Passman (for example Gonçalves and Passman, 2004) have looked at the construction of free products in unit groups of group rings of finite groups, and the present authors have considered the same question in relation to elements of finite order in central simple algebras (Shirvani and Gonçalves, 2005).…”

Section: Introductionmentioning

confidence: 97%