VECTOR SPACE GENERATED BY THE MULTIPLICATIVE COMMUTATORS OF A DIVISION RING

2013 ◽  
Vol 12 (08) ◽  
pp. 1350043 ◽  
Author(s):  
M. AGHABALI ◽  
S. AKBARI ◽  
M. ARIANNEJAD ◽  
A. MADADI
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Characteristic Zero

Let D be a division ring with center F. An element of the form xyx-1y-1 ∈ D is called a multiplicative commutator. Let T(D) be the vector space over F generated by all multiplicative commutators in D. In this paper it is shown that if D is algebraic over F and Char (D) = 0, then D = T(D). We conjecture that it is true in general. Among other results it is shown that in characteristic zero if T(D) is algebraic over F, then D is algebraic over F.

Finite quasisimple groups of 2×2 matrices over a division ring

1985 ◽  
Vol 97 (3) ◽  
pp. 415-420
Author(s):  
B. Hartley ◽  
M. A. Shahabi Shojaei
Keyword(s):  
Finite Groups ◽  
Division Ring ◽  
Characteristic Zero ◽  
Multiplicative Group ◽  
Main Tool ◽  
Number Fields ◽  
Soluble Groups ◽  
Trivial Element ◽  
Difficult Part

In 1955 [1], Amitsur determined all the finite groups G that can be embedded in the multiplicative group T* = GL(1, T) of some division ring T of characteristic zero. If G can be so embedded, then the rational span of G in T is a division ring of finite dimension over ℚ, and G acts on it by right multiplication in such a way that every non-trivial element operates fixed point freely. The finite groups admitting such a representation had earlier been determined by Zassenhaus[24; 4, XII. 8], and Amitsur begins by quoting Zassenhaus' results, which show in particular that the only perfect group that can be embedded in the multiplicative group of a division ring of characteristic zero is SL(2,5). The more difficult part of Amitsur's paper is the determination of the possible soluble groups. Here the main tool is Hasse's theory of cyclic algebras over number fields.

scholarly journals On subgroups in division rings of type 2

2012 ◽  
Vol 49 (4) ◽  
pp. 549-557
Author(s):  
Bui Hai ◽  
Trinh Deo ◽  
Mai Bien
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Division Rings ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Multiplicative Subgroups

Let D be a division ring with center F. We say that D is a division ring of type 2 if for every two elements x, y ∈ D, the division subring F(x, y) is a finite dimensional vector space over F. In this paper we investigate multiplicative subgroups in such a ring.

scholarly journals A note on multiplicative commutators of division rings

2019 ◽  
Vol 18 (02) ◽  
pp. 1950031
Author(s):  
Roozbeh Hazrat
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Commutator Subgroup ◽  
Division Rings

We give an example of a division ring [Formula: see text] whose multiplicative commutator subgroup does not generate [Formula: see text] as a vector space over its center, thus disproving the conjecture posed in [M. Aghabali, S. Akbari, M. Ariannejad and A. Madadi, Vector space generated by the multiplicative commutators of a division ring, J. Algebra Appl. 12(8) (2013) 7 pp.].

scholarly journals NILPOTENT MODULES OVER POLYNOMIAL RINGS

2020 ◽  
pp. 20-25
Author(s):  
А. Petravchuk ◽  
Ie. Chapovskyi ◽  
I. Klimenko ◽  
M. Sidorov
Keyword(s):  
Vector Space ◽  
Polynomial Ring ◽  
Characteristic Zero ◽  
Finite Dimension ◽  
Polynomial Rings ◽  
Closed Field ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Groups Of Automorphisms

Let K be an algebra ically closed field of characteristic zero, K[X ] the polynomial ring in n variables. The vector space Tn = K[X] is a K[X ] -module with the action i = xi 'x  v v for vTn . Every finite dimensional submodule V of Tn is nilpotent, i.e. every f  K[X ] acts nilpotently (by multiplication) on V . We prove that every nilpotent K[X ] -module V of finite dimension over K with one-dimensional socle can be isomorphically embedded in the module Tn . The groups of automorphisms of the module Tn and its finite dimensional monomial submodules are found. Similar results are obtained for (non-nilpotent) finite dimensional K[X ] -modules with one dimensional socle.

scholarly journals LINEAR DIVISION RINGS

2009 ◽  
Vol 12 (17) ◽  
pp. 5-11
Author(s):  
Bien Hoang Mai ◽  
Hai Xuan Bui
Keyword(s):  
Vector Space ◽  
Division Ring ◽  
Finite Subset ◽  
Multiplicative Group ◽  
Dimensional Vector ◽  
Finitely Generated ◽  
Dimensional Vector Space ◽  
Division Rings ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

Let D be a division ring with the center F and suppose that D* is the multiplicative group of D. D is called centrally finite if D is a finite dimensional vector space over F and D is locally centrally finite if every finite subset of D generates over F a division subring which is a finite dimensional vector space over F. We say that D is a linear division ring if every finite subset of D generates over Fa centrally finite division subring. It is obvious that every locally centrally finite division ring is linear. In this report we show that the inverse is not true by giving an example of a linear division ring which is not locally centrally finite. Further, we give some properties of subgroups in linear division rings. In particular, we show that every finitely generated subnormal subgroup in a linear ring is central. An interesting corollary is obtained as the following: If D is a linear division ring and D* is finitely generated, then D is a finite field.

scholarly journals A p-adic Montel Theorem and locally polynomial functions

Filomat ◽  
2014 ◽  
Vol 28 (1) ◽  
pp. 159-166
Author(s):  
J.M. Almira ◽  
Kh.F. Abu-Helaiel
Keyword(s):  
Functional Equation ◽  
General Solution ◽  
Vector Space ◽  
Normed Space ◽  
Characteristic Zero ◽  
Continuous Functions ◽  
Polynomial Functions ◽  
Valued Field ◽  
Q Vector

We prove a version of Montel?s Theorem for the case of continuous functions defined over the field Qp of p-adic numbers. In particular, we prove that, if ?m+1 h0 f (x) = 0 for all x ? Qp, and h0 satisfies |h0|p = p?N0, then, for all x0 ? Qp, the restriction of f over the set x0 + pN0Zp coincides with a polynomial px0 (x) = a0(x0) + a1(x0)x +...+ am(x0)xm. Motivated by this result, we compute the general solution of the functional equation with restrictions given by ?m+1 h f (x) = 0 (x ? X and h ? BX(r) = {x ? X : ?x? ? r}), whenever f : X ? Y, X is an ultrametric normed space over a non-Archimedean valued field (K, |?|) of characteristic zero, and Y is a Q-vector space. By obvious reasons, we call these functions uniformly locally polynomial.

scholarly journals Free symmetric algebras in division rings generated by enveloping algebras of Lie algebras

2015 ◽  
Vol 25 (06) ◽  
pp. 1075-1106 ◽  
Author(s):  
Vitor O. Ferreira ◽  
Jairo Z. Gonçalves ◽  
Javier Sánchez
Keyword(s):  
Large Class ◽  
Lie Algebras ◽  
Division Ring ◽  
Characteristic Zero ◽  
Free Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Symmetric Algebras ◽  
Ring Of Fractions ◽  
Ore Domain

For any Lie algebra L over a field, its universal enveloping algebra U(L) can be embedded in a division ring 𝔇(L) constructed by Lichtman. If U(L) is an Ore domain, 𝔇(L) coincides with its ring of fractions. It is well known that the principal involution of L, x ↦ -x, can be extended to an involution of U(L), and Cimpric proved that this involution can be extended to one on 𝔇(L). For a large class of noncommutative Lie algebras L over a field of characteristic zero, we show that 𝔇(L) contains noncommutative free algebras generated by symmetric elements with respect to (the extension of) the principal involution. This class contains all noncommutative Lie algebras such that U(L) is an Ore domain.

scholarly journals Invariants of Finite Reflection Groups

1963 ◽  
Vol 22 ◽  
pp. 57-64 ◽  
Author(s):  
Louis Solomon
Keyword(s):  
Vector Space ◽  
Characteristic Zero ◽  
Polynomial Functions ◽  
Dimensional Vector ◽  
Reflection Groups ◽  
Linear Transformations ◽  
Graded Ring ◽  
Polynomial Invariants ◽  
Dimensional Vector Space ◽  
Group Of Automorphisms

Let K be a field of characteristic zero. Let V be an n-dimensional vector space over K and let S be the graded ring of polynomial functions on V. If G is a group of linear transformations of V, then G acts naturally as a group of automorphisms of S if we defineThe elements of S invariant under all γ ∈ G constitute a homogeneous subring I(S) of S called the ring of polynomial invariants of G.

Strongly and co-strongly minimal abelian structures

2010 ◽  
Vol 75 (2) ◽  
pp. 442-458 ◽  
Author(s):  
Ehud Hrushovski ◽  
James Loveys
Keyword(s):  
Vector Space ◽  
Finite Index ◽  
Division Ring ◽  
First Case ◽  
Special Cases ◽  
Structure Theorems ◽  
Algebraic Element ◽  
Minimal Module ◽  
Induced Structure

AbstractWe give several characterizations of weakly minimal abelian structures. In two special cases, dual in a sense to be made explicit below, we give precise structure theorems:1. when the only finite 0-definable subgroup is {0}, or equivalently 0 is the only algebraic element (the co-strongly minimal case);2. when the theory of the structure is strongly minimal.In the first case, we identify the abelian structure as a “near-subspace” A of a vector space V over a division ring D with its induced structure, with possibly some collection of distinguished subgroups of A of finite index in A and (up to acl(∅)) no further structure. In the second, the structure is that of V/A for a vector space and near-subspace as above, with the only further possible structure some collection of distinguished points. Here a near-subspace of V is a subgroup A such that for any nonzero d ∈ D. the index of A ∩ dA, in A is finite. We also show that any weakly minimal abelian structure is a reduct of a weakly minimal module.

The Second Cohomology of Current Algebras of General Lie Algebras

2008 ◽  
Vol 60 (4) ◽  
pp. 892-922 ◽  
Author(s):  
Karl-Hermann Neeb ◽  
Friedrich Wagemann
Keyword(s):  
Vector Space ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Characteristic Zero ◽  
Cohomology Space ◽  
Trivial Module ◽  
Commutative Associative Algebra ◽  
Locally Convex ◽  
Algebra Over A Field

AbstractLet A be a unital commutative associative algebra over a field of characteristic zero, a Lie algebra, and a vector space, considered as a trivial module of the Lie algebra . In this paper, we give a description of the cohomology space in terms of easily accessible data associated with A and . We also discuss the topological situation, where A and are locally convex algebras.

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