Invariance conditions on substructures of division rings

Aaghabali, Mehdi; Amiri, Mohsen; Ariannejad, M.; Madadi, A.

doi:10.1142/s0219498816500584

Cited by 3 publications

(1 citation statement)

References 8 publications

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“…Bois and G. Vernik in [9, Theorem 2.1.5] proved that if F is a field with Char(F ) = p > 2 and L is a finite dimensional non-abelian solvable Lie algebra over F, then K(L), the division ring of fractions of the enveloping algebra U (L), contains some purely inseparable maximal subfield. There are more results regarding self-invariant subfields of division rings in [1].…”

Section: Introductionmentioning

confidence: 99%

Self-Invariant Maximal Subfields and Their Connexion with Some Conjectures in Division Rings

Aaghabali¹,

Bien²

2019

Preprint

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Let D be a division algebra with center F . A maximal subfield of D is defined to be a fieldThis kind of subfields is important because they have strong connexion with most famous Albert's Conjecture (every division ring of prime index is cyclic). In fact, we pose a question that asserts whether every division ring whose all maximal subfields are self-invariant has to be commutative. The positive answer to this question, in finite dimensional case, implies the Albert's Conjecture (see §2). Although we show the Mal'cev-Neumann division ring demonstrates negative answer in the case of infinite dimensional division rings, but it is still most likely the question receives positive answer if we restrict ourselves to the finite dimensional division rings. We also have had the opportunity to use the Mal'cev-Neumann structure to answer Conjecture 1 below in negative (see §3). Finally, among other things, we rely on this kind of subfields to present a criteria for a division ring to have finite dimensional subdivision ring (see §4).

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Section: Introductionmentioning

confidence: 99%

Self-Invariant Maximal Subfields and Their Connexion with Some Conjectures in Division Rings

Aaghabali¹,

Bien²

2019

Preprint

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Subnormal subgroups and self-invariant maximal subfields in division rings

Aaghabali

Bien

2021

Journal of Algebra

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Conditions on Lie ideals in rings

et al. 2016

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We show that if [Formula: see text] is a non-central Lie ideal of a ring [Formula: see text] with Char[Formula: see text], such that all of its nonzero elements are invertible, then [Formula: see text] is a division ring. We prove that if [Formula: see text] is an [Formula: see text]-central algebra and [Formula: see text] is a Lie ideal without zero divisor such that the set of multiplicative cosets [Formula: see text] is of finite cardinality, then either [Formula: see text] is a field or [Formula: see text] is central. We show the only non-central Lie ideal without zero divisor of a non-commutative central [Formula: see text]-algebra [Formula: see text] with Char[Formula: see text] and radical over the center is [Formula: see text], the additive commutator subgroup of [Formula: see text] and in this case [Formula: see text] is a generalized quaternion algebra. Finally we prove that if [Formula: see text] is a Lie ideal without zero divisor in a central [Formula: see text]-algebra with characteristic not 2 and if [Formula: see text] is a finite residual group, then [Formula: see text] is central.

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Invariance conditions on substructures of division rings

Cited by 3 publications

References 8 publications

Self-Invariant Maximal Subfields and Their Connexion with Some Conjectures in Division Rings

Self-Invariant Maximal Subfields and Their Connexion with Some Conjectures in Division Rings

Subnormal subgroups and self-invariant maximal subfields in division rings

Conditions on Lie ideals in rings

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