Search

Dickson’s commutative semifields are an important class of finite division algebras. We generalize Dickson’s construction over any base field by doubling not just finite field extensions (which wou...

This article investigates the cohomological kernels ker(H2n+1F→H2n+1E) of field extensions E/F in characteristic two where separable part of E is a quadratic extension F(α) and E is quadratic or qu...

Let k be a field of characteristic different from 2 and 3. In this paper, we study the number of rank-2 k-tori required (in fact exhibit such tori explicitly) for the generation of groups of type A2, G2 and F4 arising from division algebras and subgroups of type D4 of Aut(A), for A an Albert division algebra, over infinite perfect fields.

The purpose of this paper is to investigate the eight-dimensional non-associative unital division algebras {0}≠A over finite fields, admitting elementary abelian 2-group of automorphisms E≤Aut(A), ...

Let n≥3 be a natural number. Let Mn(𝔻) be the ring of all n×n matrices over a noncommutative division ring 𝔻. In the present paper, we will find the description of all additive mappings G:Mn(𝔻)→Mn(𝔻) such that [G(y),y] = G(y)y−yG(y) = 0 for all rank-1 matrix y. Precisely, we will prove that G(x) = λx+μ(x) for all x∈Mn(𝔻), where λ lies in the center of 𝔻 and μ is a central map.

Let F be a Henselian field such that F¯ is a global field. It is shown that if D is an F-central division algebra such that charF¯∤deg(D) then D contains a cyclic division algebra of prime degree even though D can be a noncrossed product. Here, our approach is based on the Albert’s cyclicity criterion and centralizer theorem.

Let k be a field of characteristic different from 2 and K a noncommutative division k-algebra of finite dimension over k. In the paper, subgroups of the general linear group GL2(K) that contain a subgroup which is conjugate in GL2(K) to the group of matrices with r ∈ k are described.

Among other results of this paper, we single out the following ones. We show that division rings are the only division semirings over which the categories of semimodules are Schreier varieties, i.e., all subsemimodules of free semimodules are free too. We a complete description of division semirings R over which the categories of semimodules ℳ R are p-Schreier varieties, i.e., varieties whose all projective algebras are free. We give a complete description of proper division semirings R whose categories of semimodules ℳ R(X) over the polynomial semirings R(X) over R, in not necessary commuting variables X, are p-Schreier varieties. We show that the categories of semimodules ℳ R(X) over the polynomial semirings R(X) over N-valued semirings R, in particular ℳ N(X), are p-Schreier varieties. We also show that for N-valued semirings S, the semimodule categories ℳ S never are Schreier varieties.

In this article, we consider categories of all semimodules over semirings which are p-Schreier varieties, i.e., varieties whose projective algebras are all free. Among other results, we show that over a division semiring R all semimodules are projective iff R is a division ring, prove that categories of all semimodules over proper additively π-regular semirings are not p-Schreier varieties (in particular, this result solves Problem 1 of Katsov [8]), as well as prove that categories of all semimodules over cancellative division semirings are, in contrast, p-Schreier varieties.

We show in this article that in many cases the subfields of a nondegenerate tame semiramified division algebra of prime power degree over a Henselian valued field are inertial field extensions of t...

The main goal of this article is to give examples of division p-algebras that are not tensor product of cyclic algebras (Corollary 2.19) and to prove that nondegenerate tame semiramified division algebras of prime power degree over a Henselian valued field are indecomposable (Theorem 3.5). For this, we give new results concerning nicely semiramified division algebras over Henselian valued fields, and we develop a new study for nondegenerate valued and graded division algebras.

Left distributive left quasigroups are binary algebras with unique left division satisfying the left distributive identity x(yz) ≈ (xy)(xz). In other words, binary algebras where all left translations are automorphisms. We provide a description and examples of nonidempotent subdirectly irreducible algebras in this class.

Given rings $R \subseteq S$, consider the division closure $DC(R,S)$ and the rational closure $RC(R,S)$ of R in S. If S is commutative, then $DC(R,S)=RC(R,S)=RT^{-1}$, where $T = \{t\in R : t^{-1} \in S\}$. We show that this is also true if we assume only that R is commutative., 5 pages; minor revisions upon referee's suggestions

We study central extensions of division rings with orderings of higher level. We show that orderings extend to certain immediate and inert extensions. Further, it is proven that any n-ordered division ring can be extended to a n-ordered division ring with almost real closed center. In the last section, an old theorem due to Neumann is generalized: every n-ordered division ring can be extended to a n-ordered division ring containing ℝ in its center.

In this paper, it is proved that if F is a global field, then for any integer n > 3, there is an extension field E over F of degree n such that K 2 E is not generated by the Steinberg symbols {a, b} with a ∈ F*, b, ∈ E*. If however, F is a number field and D is a finite-dimensional central division F-algebra with square free index, then K 2 D is always generated by the Steinberg symbols {a, b} with a ∈ F*, b ∈ D*. Finally, the tame kernels of central division algebras over F are expressed explicitly.

Let G be a locally indicable group, K a division ring, and KG a crossed-product group ring. In 1961, Ian Hughes proved that, up to KG-isomorphism, at most one division ring of fractions of KG satisfies a certain independence condition, now called Hughes freeness. This result was applied by others in work on division rings of fractions of group rings of free groups. In this article, we introduce concepts that illuminate Hughes' arguments, and we simplify the proof of the theorem.

Among the algebraic identities satisfied by alternative algebras play a fundamental role the so called Moufang identities. In the first section of this note we characterize division algebras satisfying one of the sided Moufang identities. As a consequence we obtain all the division normed real Moufang sided algebras. Up to isomorphism there are seven algebras in both cases, including the well known alternative algebras , , and . Moreover, there are three division right Moufang algebras with dimensions 2, 4 and 8 and another three division left Moufang algebras of the same dimensions. Finally we prove that , , and are the only division normed algebras satisfying the Moufang middle identity.

In the paper we determine all valuation on division rings of quantum polynomial rings (quantum fields). Furthermore we consider completions of quantum fields with respect to maximal valuations and study their automorphism groups and derivations.

In [6] D. W. Lewis proved the existence of a quaternion division algebra of level 2 n or 2 n + 1 for any integer n ≥ 1. In this note, we present news examples of quaternion division algebras of such levels. We also recover a result of P. Koprowski [4]. Our method is different and based on function field of projective quadrics.

In this paper we present a division theorem for the pseudovariety of semigroups OP generated by all semigroups of orientation preserving full transformations on a finite chain, which is achieved by using a natural representation of a certain monoid of transformations of a set X as a monoid of transformations of a subset of X.

Let k ⊆ L be division rings, with [L: k]right

(1998). Algebre a division graduee cent rale. Communications in Algebra: Vol. 26, No. 9, pp. 2933-2947.

In this paper we study division rings D in which the elements of the derived group D1 of D∗ = D - {0} are algebraic over the centre of D.

The aim of this paper is to prove that, except in caracteristic 2, the division rings considered in H. Lorenz's paper [1] do not only contain free associative algebras, but also free group algebras. The main ideas are due to L. Makar-Limanov who proved that result for D1 [2]

In this paper we construct Baer ordered indecomposable division algebras of index p n and exponent p m for all primes p and for all n m, with n>m≥1 (n≥3 if p=2).

We consider Euclidean domains and their groups of units. Let K(a,b) be the set of remainders in the division of a by b. If Card K(a,b) = 1 for any a and b from a Euclidean domain R, then R is known to be isomorphic to the ring of polynomials over some field, see [4], [5]. On the other hand, the condition Card K(a,b) = 2 for any a and b implies that R is isomorphic to the ring Z of integers, see [2]. We give characterization of Euclidean domains and their groups of units under some other conditions on K(a,b).

A semisimple Jordan pair has dcc on inner ideals if and only if it is covered by a finite grid whose simultaneous Peirce spaces are division pairs. The proof uses a characterization of inner ideals in terms of grids which follows from a conjugacy theorem for grids.