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]).

Antiautomorphisms with quasi-generalized Engel condition

2018 ◽  
Vol 17 (08) ◽  
pp. 1850145 ◽  
Author(s):  
M. Chacron
Keyword(s):  
Finite Order ◽  
Prime Ring ◽  
Division Ring ◽  
Idempotent Element ◽  
General Question ◽  
Engel Condition ◽  
Positive Integers

Let [Formula: see text] be a ring with 1. Given elements [Formula: see text], [Formula: see text] of [Formula: see text] and the integer [Formula: see text] define [Formula: see text] and [Formula: see text]. We say that a given antiautomorphism [Formula: see text] of [Formula: see text] is commuting if [Formula: see text], all [Formula: see text]. More generally, assume that [Formula: see text] satisfies the condition [Formula: see text] where [Formula: see text], [Formula: see text] are corresponding positive integers depending on [Formula: see text], and [Formula: see text] ranges over [Formula: see text]. To what extent can one say that [Formula: see text] is commuting? In this paper, we answer the question in the affirmative if R is a prime ring containing some idempotent element [Formula: see text]. In the diametrically opposed case in which [Formula: see text] is a division ring the answer is again yes provided [Formula: see text] is algebraic over its center and [Formula: see text] is of finite order. These two major complementary results will be put to work to provide an answer to the general question.

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 question argument

2018 ◽  
Author(s):  
Tristram McPherson
Keyword(s):  
Natural Sciences ◽  
Ethical Naturalism ◽  
Natural Properties ◽  
The Core ◽  
Open Questions ◽  
Open Question Argument ◽  
Naturalistic Account ◽  
Open Question

The open question argument is the heart of G.E. Moore’s case against ethical naturalism. Ethical naturalism is the view that goodness, rightness, etc. are natural properties; roughly, the sorts of properties that can be investigated by the natural sciences. Moore claims that, for any candidate naturalistic account of an ethical term according to which ‘good’ had the same meaning as some naturalistic term A, we might without confusion ask: ‘I see that this act is A, but is it good?’ Moore claimed that the existence of such open questions shows that ethical naturalism is mistaken. In the century since its introduction, the open question argument has faced a battery of objections. Despite these challenges, some contemporary philosophers claim that the core of Moore’s argument can be salvaged. The most influential defences link Moore’s argument to the difficulty that naturalistic ethical realists face in explaining the practical role of ethical concepts in deliberation.

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.

AN ENGEL CONDITION WITH AUTOMORPHISMS FOR LEFT IDEALS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350092 ◽  
Author(s):  
CHENG-KAI LIU
Keyword(s):  
Left Ideal ◽  
Prime Ring ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Identity Automorphism ◽  
Semiprime Rings ◽  
Engel Condition ◽  
Positive Integers

Let R be a prime ring and L a nonzero left ideal of R. For x, y ∈ R, we denote [x, y] = xy-yx the commutator of x and y. In this paper, we prove that if R admits a non-identity automorphism σ such that [[…[[σ(xn0), xn1], xn2], …], xnk] = 0 for all x ∈ L, where n0, n1, n2, …, nk are fixed positive integers, then R is commutative. The analogous results for semiprime rings and von Neumann algebras are also obtained.

DERIVATIONS OF FRÉCHET NUCLEAR GB-ALGEBRAS

2015 ◽  
Vol 92 (2) ◽  
pp. 290-301 ◽  
Author(s):  
M. WEIGT ◽  
I. ZARAKAS
Keyword(s):  
Affirmative Answer ◽  
Open Question

It is an open question whether every derivation of a Fréchet GB$^{\ast }$-algebra $A[{\it\tau}]$ is continuous. We give an affirmative answer for the case where $A[{\it\tau}]$ is a smooth Fréchet nuclear GB$^{\ast }$-algebra. Motivated by this result, we give examples of smooth Fréchet nuclear GB$^{\ast }$-algebras which are not pro-C$^{\ast }$-algebras.

Extensions and applications of a tauberian theorem due to valiron

1951 ◽  
Vol 47 (1) ◽  
pp. 22-37 ◽  
Author(s):  
M. E. Noble
Keyword(s):  
Finite Order ◽  
Tauberian Theorem ◽  
Positive Integers

1. Introduction. In the course of an important memoir on integral functions of finite order†, G. Valiron discusses ‘fonctions orientées’, that is, functions with zeros an such that arg an tends to a limit as n → ∞. He obtains results that include the following two theorems, in which Vρ(x) denotes a function of the form , where α1, …, αν are positive integers, and n(r) has its usual significance in the theory of integral functions:.

scholarly journals An extension of the Kegel–Wielandt theorem to locally finite groups

1996 ◽  
Vol 38 (2) ◽  
pp. 171-176
Author(s):  
Silvana Franciosi ◽  
Francesco de Giovanni ◽  
Yaroslav P. Sysak
Keyword(s):  
Finite Group ◽  
Affirmative Answer ◽  
Locally Finite Group ◽  
Linear Groups ◽  
Arbitrary Group ◽  
Famous Theorem ◽  
Locally Finite ◽  
Locally Nilpotent ◽  
Finite Products ◽  
Open Question

A famous theorem of Kegel and Wielandt states that every finite group which is the product of two nilpotent subgroups is soluble (see [1], Theorem 2.4.3). On the other hand, it is an open question whether an arbitrary group factorized by two nilpotent subgroups satisfies some solubility condition, and only a few partial results are known on this subject. In particular, Kegel [6] obtained an affirmative answer in the case of linear groups, and in the same article he also proved that every locally finite group which is the product of two locally nilpotent FC-subgroups is locally soluble. Recall that a group G is said to be an FC-group if every element of G has only finitely many conjugates. Moreover, Kazarin [5] showed that if the locally finite group G = AB is factorized by an abelian subgroup A and a locally nilpotent subgroup B, then G is locally soluble. The aim of this article is to prove the following extension of the Kegel–Wielandt theorem to locally finite products of hypercentral groups.

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).

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