Kummer subfields of tame division algebras over Henselian fields

By generalizing the method used by Tignol and Amitsur in [J.-P. Tignol, S.A. Amitsur, Kummer subfields of Malcev-Neumann division algebras, Israel Journal of Math. 50 (1985), 114-144], we determine necessary and sufficient conditions for an arbitrary central division algebra D over a Henselian valued field E to have Kummer subfields when the characteristic of the residue field of E does not divide the degree of D. We prove also that if D is a semiramified division algebra of degree n [resp., of prime power degree p^r] over E such that does not divide n and [resp., and p³ divides ], then D is non-cyclic [resp., D is not an elementary abelian crossed product].     

Decomposability for division algebras of exponent two and associated forms

This article investigates the structure of quadratic forms and of division algebras of exponent two over fields of characteristic different from two with the property that the third power of the fundamental ideal in the associated Witt ring is torsion free.      

Decomposability of free Tarski algebras

In this paper we study the direct decomposability of free Tarski algebras. We show that infinite freely generated Tarski algebras are directly indecomposable, whereas finite freely generated Tarski algebras can only be decomposed into a direct product of two factors, one of which is the two-element Tarski algebra.     

Universal state space embeddability of Jordan-Banach algebras

We study extensions of states between projection structures of JB algebras and generalized orthomodular posets. It is shown that projection orthoposet of a JB algebra admits the universal extension property if and only if the Gleason theorem is valid for . As a consequence we get that any positive Stone algebra-valued measure on projection lattice of a quotient of a JBW algebra without type direct summand extends to a positive measure on an arbitrary larger generalized orthomodular lattice.

     

Decomposability of free Łukasiewicz implication algebras

?ukasiewicz implication algebras are {→,1}-subreducts of Wajsberg algebras (MV-algebras). They are the algebraic counterpart of Super-?ukasiewicz Implicational logics investigated in Komori, Nogoya Math J 72:127–133, 1978. The aim of this paper is to study the direct decomposability of free ?ukasiewicz implication algebras. We show that freely generated algebras are directly indecomposable. We also study the direct decomposability in free algebras of all its proper subvarieties and show that infinitely freely generated algebras are indecomposable, while finitely free generated algebras can be only decomposed into a direct product of two factors, one of which is the two-element implication algebra.     

Completion and finite embeddability property for residuated ordered algebras.

A residuated ordered algebra is a partially ordered set with additional ‘residuated’ operations. A construction is presented that, from any partial subalgebra of a residuated ordered algebra, constructs a complete algebra into which the partial subalgebra embeds. Conditions are given under which the constructed algebra is finite whenever a finite partial subalgebra is chosen. This implies the ‘finite embeddability property’ for the given class of residuated ordered algebras. In the case that the whole algebra is chosen as the partial subalgebra, the construction is a completion of the underlying order of the algebra. A scheme of inequalities is described that are shown to have the property of being preserved by the above construction. These preservation results thus extend the results on the finite embeddability property and completion.     

Graded division algebras

Mathematische Zeitschrift -     

Cyclic division algebras

A general example of cyclic division algebra is given, based on a construction of Brauer, yielding examples of division algebras of arbitrary prime exponent without proper central subalgebras, and also noncrossed products of arbitrary exponent. This research was supported in part by the U.S.-Israel Binational Science Foundation. An erratum to this article is available at .     

Selectivity in division algebras

A commutative order in a central simple algebra over a number field is said to be selective if it embeds in some, but not all, maximal orders in the algebra. We completely characterize selective orders in central division algebras, of dimension 9 or greater, in terms of the characterization of selective orders given by Chinburg and Friedman in the quaternionic case.     

On central division algebras

For any integern such that 8|n or for which there exists an odd primeq such thatq ²|n, there is a central division algebra of dimensionn ² over its center which is not a crossed product. The algebra constructed in this paper is the algebraQ(X ₁,…,X)_m, the algebra generated over the rationalQ bym(≧2) generic matrices. To the memory of A. A. Albert This paper was originally presented in November, 1971 for publication elsewhere in a volume in honor of Prof. A. A. Albert on the occasion of his 65th birthday. The volume was never published due to the death of Prof. Albert in June 1972.     

Semidirect product division algebras

SupposeD is a division algebra of degreep over its centerF, which contains a primitivep-root of 1. Also supposeD has a maximal separable subfield overF whose Galois group is the semidirect product of the cyclic groupsC _p C _q , whereq=2, 3, 4, or 6 and is relatively prime top (In particular this is the case whenp is prime ≤7 andD has a maximal separable subfield whose Galois group is solvable.) ThenD is cyclic. The proof involves developing a theory of a wider class of algebras, which we call accessible, and proving that they are cyclic.