scholarly journals The universality of Hughes-free division rings

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Andrei Jaikin-Zapirain
Keyword(s):  
Division Ring ◽  
Amenable Group ◽  
Crossed Product ◽  
Division Rings ◽  
Ring Of Fractions ◽  
Open Question ◽  
Indicable Group

AbstractLet $$E*G$$ E ∗ G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to $$E*G$$ E ∗ G -isomorphism, there exists at most one Hughes-free division $$E*G$$ E ∗ G -ring. However, the existence of a Hughes-free division $$E*G$$ E ∗ G -ring $${\mathcal {D}}_{E*G}$$ D E ∗ G for an arbitrary locally indicable group G is still an open question. Nevertheless, $${\mathcal {D}}_{E*G}$$ D E ∗ G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether $${\mathcal {D}}_{E*G}$$ D E ∗ G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists $${\mathcal {D}}_{E[G]}$$ D E [ G ] and it is universal. In Appendix we give a description of $${\mathcal {D}}_{E[G]}$$ D E [ G ] when G is a RFRS group.

Open questions concerning antiautomorphisms of division rings with quasi-generalized Engel conditions

2019 ◽  
Vol 18 (09) ◽  
pp. 1950167 ◽  
Author(s):  
M. Chacron ◽  
T.-K. Lee
Keyword(s):  
Finite Order ◽  
Division Ring ◽  
Affirmative Answer ◽  
Major Result ◽  
Division Rings ◽  
Open Questions ◽  
Engel Condition ◽  
Open Question ◽  
Positive Integers

Let [Formula: see text] be a noncommutative division ring with center [Formula: see text], which is algebraic, that is, [Formula: see text] is an algebraic algebra over the field [Formula: see text]. Let [Formula: see text] be an antiautomorphism of [Formula: see text] such that (i) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are positive integers depending on [Formula: see text]. If, further, [Formula: see text] has finite order, it was shown in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] that [Formula: see text] is commuting, that is, [Formula: see text], all [Formula: see text]. Posed in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] is the question which asks as to whether the finite order requirement on [Formula: see text] can be dropped. We provide here an affirmative answer to the question. The second major result of this paper is concerned with a nonnecessarily algebraic division ring [Formula: see text] with an antiautomorphism [Formula: see text] satisfying the stronger condition (ii) [Formula: see text], all [Formula: see text], where [Formula: see text] and [Formula: see text] are fixed positive integers. It was shown in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036] that if, further, [Formula: see text] has finite order then [Formula: see text] is commuting. We show here, that again the finite order assumption on [Formula: see text] can be lifted answering thus in the affirmative the open question (see Question 2.11 in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036]).

scholarly journals Free division rings of fractions of crossed products of groups with Conradian left-orders

2020 ◽  
Vol 32 (3) ◽  
pp. 739-772
Author(s):  
Joachim Gräter
Keyword(s):  
Vector Space ◽  
Endomorphism Ring ◽  
Division Ring ◽  
Skew Field ◽  
Division Rings ◽  
Products Of Groups ◽  
Ring Of Fractions ◽  
Left Order ◽  
The Right ◽  
Formal Power

AbstractLet D be a division ring of fractions of a crossed product {F[G,\eta,\alpha]}, where F is a skew field and G is a group with Conradian left-order {\leq}. For D we introduce the notion of freeness with respect to {\leq} and show that D is free in this sense if and only if D can canonically be embedded into the endomorphism ring of the right F-vector space {F((G))} of all formal power series in G over F with respect to {\leq}. From this we obtain that all division rings of fractions of {F[G,\eta,\alpha]} which are free with respect to at least one Conradian left-order of G are isomorphic and that they are free with respect to any Conradian left-order of G. Moreover, {F[G,\eta,\alpha]} possesses a division ring of fraction which is free in this sense if and only if the rational closure of {F[G,\eta,\alpha]} in the endomorphism ring of the corresponding right F-vector space {F((G))} is a skew field.

Subnormal Subgroups of Division Rings

1963 ◽  
Vol 15 ◽  
pp. 80-83 ◽  
Author(s):  
I. N. Herstein ◽  
W. R. Scott
Keyword(s):  
Normal Subgroup ◽  
Division Ring ◽  
Multiplicative Group ◽  
Finite Sequence ◽  
Division Rings ◽  
Subnormal Subgroups

Let K be a division ring. A subgroup H of the multiplicative group K′ of K is subnormal if there is a finite sequence (H = A0, A1, . . . , An = K′) of subgroups of K′ such that each Ai is a normal subgroup of Ai+1. It is known (2, 3) that if H is a subdivision ring of K such that H′ is subnormal in K′, then either H = K or H is in the centre Z(K) of K.

Derivations with Invertible Values on a Lie Ideal

1988 ◽  
Vol 31 (1) ◽  
pp. 103-110 ◽  
Author(s):  
Jeffrey Bergen ◽  
L. Carini
Keyword(s):  
Division Ring ◽  
Unit Element ◽  
Lie Ideal ◽  
Division Rings

AbstractLet R be a ring which possesses a unit element, a Lie ideal U ⊄ Z, and a derivation d such that d(U) ≠ 0 and d(u) is 0 or invertible, for all u ∈ U. We prove that R must be either a division ring D or D2, the 2 X 2 matrices over a division ring unless d is not inner, R is not semiprime, and either 2R or 3R is 0. We also examine for which division rings D, D2 can possess such a derivation and study when this derivation must be inner.

scholarly journals Almost Finiteness for General Étale Groupoids and Its Applications to Stable Rank of Crossed Products

2018 ◽  
Vol 2020 (19) ◽  
pp. 6007-6041 ◽  
Author(s):  
Yuhei Suzuki
Keyword(s):  
Amenable Group ◽  
Cantor Set ◽  
Stable Rank ◽  
Crossed Product ◽  
Local Version ◽  
Crossed Products ◽  
Subexponential Growth ◽  
Minimal Action ◽  
New Strategy ◽  
Totally Disconnected

Abstract We extend Matui’s notion of almost finiteness to general étale groupoids and show that the reduced groupoid C$^{\ast }$-algebras of minimal almost finite groupoids have stable rank 1. The proof follows a new strategy, which can be regarded as a local version of the large subalgebra argument. The following three are the main consequences of our result: (1) for any group of (local) subexponential growth and for any its minimal action admitting a totally disconnected free factor, the crossed product has stable rank 1; (2) any countable amenable group admits a minimal action on the Cantor set, all whose minimal extensions form the crossed product of stable rank 1; and (3) for any amenable group, the crossed product of the universal minimal action has stable rank 1.

scholarly journals On the Quadratic Extensions and the Extended Witt Ring of a Commutative Ring

1973 ◽  
Vol 49 ◽  
pp. 127-141 ◽  
Author(s):  
Teruo Kanzaki
Keyword(s):  
Galois Group ◽  
Commutative Ring ◽  
Galois Extension ◽  
Identity Element ◽  
Quadratic Extension ◽  
Crossed Product ◽  
Witt Ring ◽  
Division Rings ◽  
Quadratic Extensions ◽  
The Common

Let B be a ring and A a subring of B with the common identity element 1. If the residue A-module B/A is inversible as an A-A- bimodule, i.e. B/A ⊗A HomA(B/A, A) ≈ HomA(B/A, A) ⊗A B/A ≈ A, then B is called a quadratic extension of A. In the case where B and A are division rings, this definition coincides with in P. M. Cohn [2]. We can see easily that if B is a Galois extension of A with the Galois group G of order 2, in the sense of [3], and if is a quadratic extension of A. A generalized crossed product Δ(f, A, Φ, G) of a ring A and a group G of order 2, in [4], is also a quadratic extension of A.

scholarly journals On a Crossed Product of a Division Ring

1969 ◽  
Vol 35 ◽  
pp. 47-51 ◽  
Author(s):  
Nobuo Nobusawa
Keyword(s):  
Automorphism Group ◽  
Division Ring ◽  
Crossed Product ◽  
Regular Elements ◽  
Factor Set

1. Let R and C be a ring and its center, and G an automorphism group of R of order n. By a factor set {cα,τ}, we mean a system of regular elements cα,τ (α,τ ∈ G) in C such that(1)

Homomorphisms of Matrix Semigroups over Division Rings from Dimension Two to Three

2011 ◽  
Vol 18 (03) ◽  
pp. 437-446
Author(s):  
Gregor Dolinar ◽  
Janko Marovt
Keyword(s):  
Division Ring ◽  
Multiplicative Semigroup ◽  
Division Rings ◽  
Matrix Semigroups

Let 𝔻 be an arbitrary division ring and Mn(𝔻) the multiplicative semigroup of all n × n matrices over 𝔻. We describe the general form of non-degenerate homomorphisms from M2(𝔻) to M3(𝔻).

scholarly journals Classical groups over division rings of characteristic two

1972 ◽  
Vol 7 (2) ◽  
pp. 191-226 ◽  
Author(s):  
William M. Pender ◽  
G.E. Wall
Keyword(s):  
Normal Subgroup ◽  
Quadratic Form ◽  
Division Ring ◽  
Quadratic Forms ◽  
Factor Group ◽  
Classical Groups ◽  
Isometry Groups ◽  
Division Rings ◽  
Characteristic Two

The notion of quadratic form over a field of characteristic two is extended to an arbitrary division ring of characteristic two with an involution of the first kind. The resulting isometry groups are shown to have a simple normal subgroup and the structure of the factor group is calculated. It is indicated how one may define and analyse all the classical groups in a unified manner by means of quadratic forms.

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