Finite-dimensional Division Rings Research Articles

2-Local derivations on algebras of matrix-valued functions on a compactum

The present paper is devoted to 2-local derivations. In 1997, P. Semrl introduced the notion of 2-local derivations and described 2-local derivations on the algebra B(H) of all bounded linear operators on the infinite-dimensional separable Hilbert space H. After this, a number of paper were devoted to 2-local maps on different types of rings, algebras, Banach algebras and Banach spaces. A similar description for the finite-dimensional case appeared later in the paper of S. O. Kim and J. S. Kim. Y. Lin and T. Wong described 2-local derivations on matrix algebras over a finite-dimensional division ring. Sh. A. Ayupov and K. K. Kudaybergenov suggested a new technique and have generalized the above mentioned results for arbitrary Hilbert spaces. Namely they considered 2-local derivations on the algebra B(H) of all linear bounded operators on an arbitrary Hilbert space H and proved that every 2-local derivation on B(H) is a derivation. Then there appeared several papers dealing with 2-local derivations on associative algebras. In the present paper 2-lo\-cal derivations on various algebras of infinite dimensional matrix-valued functions on a compactum are described. We develop an algebraic approach to investigation of derivations and \mbox{2-local} derivations on algebras of infinite dimensional matrix-valued functions on a compactum and prove that every such 2-local derivation is a derivation. As the main result of the paper it is established that every \mbox{2-local} derivation on a ∗-algebra C(Q,Mn(F)) or C(Q,Nn(F)), where Q is a compactum, Mn(F) is the ∗-algebra of infinite dimensional matrices over complex numbers (real numbers or quaternoins) defined in section 1, Nn(F) is the ∗-subalgebra of Mn(F) defined in section 2, is a derivation. Also we explain that the method developed in the paper can be applied to Jordan and Lie algebras of infinite dimensional matrix-valued functions on a compactum.

LINEAR RECURRING SEQUENCES OVER NONCOMMUTATIVE RINGS

Contrary to the commutative case, the set of linear recurring sequences with values in a module over a noncommutative ring is no more a module for the usual operations. We show the stability of these operations when the ring is a matrix ring or a division ring. In the case of a finite dimensional division ring over its center, we give an algorithm for the determination of a recurrence relation for the sum of two linear recurring sequences.

Finite-dimensional homogeneously simple algebras of associative type

In this paper, we describe finite-dimensional homogeneously simple algebras of associative type whose 1-component is a full matrix algebra. In addition, we prove that a finite-dimensional division ring of associative type over an algebraically closed field is isomorphic to a group algebra.

A NOTE ON 2-LOCAL MAPS

AbstractThe aim of this note is to characterize 2-local automorphisms and derivations on matrix rings over finite-dimensional division rings.

FREE SYMMETRIC AND UNITARY PAIRS IN DIVISION RINGS WITH INVOLUTION

Let D be a division ring with an involution and characteristic different from 2. Then, up to a few exceptions, D contains a pair of symmetric elements freely generating a free subgroup of its multiplicative group provided that (a) it is finite-dimensional and the center has a finite sufficiently large transcendence degree over the prime field, or (b) the center is uncountable, but not algebraically closed in D. Under conditions (a), if the involution is of the first kind, it is also shown that the unitary subgroup of the multiplicative group of D contains a free subgroup, with one exception. The methods developed are also used to exhibit free subgroups in the multiplicative group of a finite-dimensional division ring provided the center has a sufficiently large transcendence degree over its prime field.

Kaplansky conjecture in the theory of quadratic forms

This article considers the splitting properties of finite-dimensional division rings over universal splitting fields of quadratic forms. An example of a field with u-invariant equal to 6 is constructed, which contradicts Kaplansky's conjecture concerning u-invariants.

THE HASSE NORM PRINCIPLE FOR ALGEBRAIC NUMBER FIELDS

The Hasse norm principle is studied for arbitrary extensions of algebraic number fields on the basis of an interpretation of the obstruction to the Hasse principle in terms of local and global Galois groups. One of the central results is the solution of Bartels' problem on the Hasse principle for subfields of finite-dimensional division rings. Bibliography: 24 titles.

DIEUDONNÉ'S CONJECTURE ON THE STRUCTURE OF UNITARY GROUPS OVER A DIVISION RING, AND HERMITIANK-THEORY

A multiplicative theory of finite-dimensional division rings with involutions of the first kind is developed in connection with Dieudonn?'s conjecture, and the structure of arbitrary division rings over Henselian fields is studied. The unitary Whitehead group UK1(?,A) is computed for Henselian division rings A with an involution ?. Classes of division rings for which Dieudonn?'s conjecture has an affirmative answer are exhibited. Bibliography: 17 titles.

Commuting rings of simple A(k)-modules

AbstractFor the Weyl algebra A(k) and each finite dimensional division ring D over k, there exists a simple A(k)-module whose commuting ring is D.It has been known for some time that if A(k) denotes the Weyl algebra over a field k of characteristic zero, the commuting ring of a simple A(k)-module is a division algebra finite dimensional over k (see the introduction of [1]). Which division algebras actually appear? Quebbemann [1] showed that if D is a finite dimensional division algebra whose center is k, then it occurs as a commuting ring. We complete this circle of ideas by showing that any D appears: a division algebra over k appears as the commuting ring of a simple A(k)-module if and only if it is finite dimensional over k.

Even-order subgroups of finite-dimensional division rings

Let K be a field. A finite group G is called K-adequate if there exists a K-division ring D (a finite-dimensional division algebra central over K) such that G is contained in D*, the multiplicative group of nonzero elements of D. Fein and Schacher have investigated the problem of determining for which fields K there exists a noncyclic group of odd order which is K-adequate. In [5, 61 they solved this problem when K is an algebraic number field or p-local field. In this paper we do the same for noncyclic groups of even order. Our main result is that if K is an algebraic number field then there exists a noncyclic group of even order which is K-adequate. We show this is false for p-local fields and determine necessary and sufficient conditions on K for there to exist an even-order noncyclic group which is K-adequate. We adopt the notation and terminology of [6]. Recall that for any natural number TZ, cIZ denotes a primitive nth root of unity. If u and ZI are integers j3(u, V) is the highest power of u dividing v and [u, v] is the order of u modulo v. If K is an algebraic number field and y is a prime of K we denote the completion of K at y by K,, . For a finite extension L of K and a prime 7 of L dividing y we denote the relative degree of q over y by f(~/r). If E and L are local fields we denote the ramification degree from L to E by e(E/L). We assume all algebras are finite-dimensional over their centers and all fields are of characteristic zero. We use freely the classification of division algebras over global and local fields by means of Hasse invariants. The reader is referred to [3, Chap. 71 for a discussion of this material.

Embedding finite groups in rational division algebras. I

In [7] and [8] the problem of determining the finite subgroups of division rings with specified centers was investigated. In particular, we were concerned with determ~ng for which fields K every odd order finite subgroup of a finite-dimensional division ring central over K is necessarily cyclic. This problem was completely settled for K a local field in [8]. In this paper we answer this question when K is an algebraic number field. Our main result is the following: Let K be an algebraic number field. Then there exists a division ring D finite dimensional and central over K such that D* has a noncyclic subgroup of odd order if and only if K contains a primitive q-th root of unity for some odd prime q. We maintain the notation and terminology of [7J and [8]. Recall that for K a field a K-division ring D is a finite-dimensional central division algebra over K. The dimension of D over K will be denoted by ED: fcl; we use the same notation for dimension of field extensions. We say that D is E-adequate if there is an E-division ring containing I). If a finite group G is contained in the multiplicative group of a K-division ring, we say that G is K-adequate. G is a K-adequate if and only if V(G) is K-adequate where V(G) is the minimal division ring containing G; the structure of T(G) was determined by Amitsur in [2]. By an A-group we will mean a noncyclic odd order group which is a subgroup of some division ring.