scholarly journals Valuation rings in finite-dimensional division algebras

1989 ◽  
Vol 120 (1) ◽  
pp. 90-99 ◽  
Author(s):  
H.H Brungs ◽  
Joachim Gräter
Keyword(s):  
Division Algebras ◽  
Finite Dimensional ◽  
Valuation Rings

Preservation of elementary equivalence under scalar extension

1982 ◽  
Vol 47 (4) ◽  
pp. 734-738
Author(s):  
Bruce I. Rose
Keyword(s):  
Identity Element ◽  
Positive Answer ◽  
General Question ◽  
Division Algebras ◽  
Extension Field ◽  
Elementary Equivalence ◽  
Applied Model ◽  
Finite Dimensional ◽  
Positive Statements ◽  
Finite Dimensional Algebras

In this note we show that taking a scalar extension of two elementarily equivalent finite-dimensional algebras over the same field preserves elementary equivalence. The general question of whether or not tensor product preserves elementary equivalence was originally raised in [4]. In [3] Feferman relates an example of Ersov which answers the question negatively. Eklof and Olin [7] also provide a counterexample to the general question in the context of two-sorted structures. Thus the result proved below is a partial positive answer to a general question whose status has been resolved negatively. From the viewpoint of applied model theory it seems desirable to find contexts in which positive statements of preservation can be obtained. Our result does have an application; a corollary to it increases our understanding of what it means for two division algebras to be elementarily equivalent.All algebras are finite-dimensional algebras over fields. All algebras contain an identity element, but are not necessarily associative.Recall that the center of a not necessarily associative algebra A is the set of elements which commute and “associate” with all elements of A. The notion of a scalar extension is an important one in algebra. If A is an algebra over F and G is an extension field of F, then the scalar extension of A by G is the algebra A ⊗F G.

Real Flexible Division Algebras

1982 ◽  
Vol 34 (3) ◽  
pp. 550-588 ◽  
Author(s):  
Georgia M. Benkart ◽  
Daniel J. Britten ◽  
J. Marshall Osborn
Keyword(s):  
Jordan Algebra ◽  
Nonassociative Algebra ◽  
Division Algebras ◽  
Hermitian Matrices ◽  
Real Numbers ◽  
The Real ◽  
Precise Statement ◽  
Finite Dimensional ◽  
Dimension One

In this paper we classify finite-dimensional flexible division algebras over the real numbers. We show that every such algebra is either (i) commutative and of dimension one or two, (ii) a slight variant of a noncommutative Jordan algebra of degree two, or (iii) an algebra defined by putting a certain product on the 3 × 3 complex skew-Hermitian matrices of trace zero. A precise statement of this result is given at the end of this section after we have developed the necessary background and terminology. In Section 3 we show that, if one also assumes that the algebra is Lie-admissible, then the structure follows rapidly from results in [2] and [3].All algebras in this paper will be assumed to be finite-dimensional. A nonassociative algebra A is called flexible if (xy)x = x(yx) for all x, y ∈ A.

scholarly journals K-theory of finite dimensional division algebras

1978 ◽  
Vol 12 (2) ◽  
pp. 153-158 ◽  
Author(s):  
S. Green ◽  
D. Handelman ◽  
P. Roberts
Keyword(s):  
Division Algebras ◽  
Finite Dimensional ◽  
K Theory

scholarly journals Nicely semiramified division algebras over Henselian fields

2005 ◽  
Vol 2005 (4) ◽  
pp. 571-577 ◽  
Author(s):  
Karim Mounirh
Keyword(s):  
Division Algebra ◽  
Division Algebras ◽  
Central Division ◽  
Field Extensions ◽  
Finite Dimensional ◽  
Central Division Algebra ◽  
Radical Type ◽  
Maximal Subfield

This paper deals with the structure of nicely semiramified valued division algebras. We prove that any defectless finite-dimensional central division algebra over a Henselian fieldEwith an inertial maximal subfield and a totally ramified maximal subfield (not necessarily of radical type) (resp., split by inertial and totally ramified field extensions ofE) is nicely semiramified.

scholarly journals Endomorphism rings of Butler groups

1987 ◽  
Vol 42 (3) ◽  
pp. 322-329 ◽  
Author(s):  
D. M. Arnold ◽  
C. I. Vinsonhaler
Keyword(s):  
Additive Group ◽  
Division Algebras ◽  
Endomorphism Rings ◽  
Dimensional Algebra ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Butler Group ◽  
Group A ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractThis note is devoted to the question of deciding whether or not a subring of a finite-dimensional algebra over the rationals, with additive group a Butler group, is the endomorphism ring of a Butler group (a Butler group is a pure subgroup of a finite direct sum of rank-1 torsion-free abelian groups). A complete answer is given for subrings of division algebras. Several applications are included.

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