Weakly isotropic quadratic forms under field extensions

For a field F of characteristic not 2, let ${\widehat{F}}$ denote the maximal dimension of anisotropic, weakly isotropic, non-degenerate quadratic forms over F. In this paper, we investigate the behavior of this invariant under field extensions.     

Transfer of quadratic forms and of quaternion algebras over quadratic field extensions

Two different proofs are given showing that a quaternion algebra Q defined over a quadratic étale extension K of a given field has a corestriction that is not a division algebra if and only if Q contains a quadratic algebra that is linearly disjoint from K. This is known in the case of a quadratic field extension in characteristic different from two. In the case where K is split, the statement recovers a well-known result on biquaternion algebras due to Albert and Draxl.     

The weak isotropy of quadratic forms over field extensions

The weak isotropy index (or equivalently, sublevel) of arbitrary quadratic forms is studied. Its relationship to the level of a form is investigated. The problem of determining the set of values of the weak isotropy index of a form as it ranges over field extensions is addressed, with both admissible and inadmissible numbers being determined. An analogous investigation with respect to the level of a form is also undertaken. A treatment of forms for which the above invariants coincide concludes this article, with some recently-raised questions being resolved.     

A matrix problem over a central quadratic skew field extension

Let F⊂G=F(u) be a central quadratic skew field extension (such that the generator u is central in G) and a natural (G,G)-bimodule. We deal with the matrix problem on finding a canonical form for rectangular matrices over W with help of left elementary transformations of their rows and right elementary transformations of columns over G. We solve this problem reducing it in the separable (resp. inseparable) case to the semilinear (resp. pseudolinear) pencil problem.     

Supersimplicity and quadratic extensions

An elliptic curve over a supersimple field with exactly one extension of degree 2 has an s-generic point. A. Martin-Pizarro would like to express his gratitude to the organisation of the Florida Logic year, in which he took part. Martin-Pizarro research was supported by a DFG-Forschungsstipendium MA3310/1-1. F. O. Wagner is a Membre junior de l’Institut Universitaire de France. Work done during the semester on Model Theory and Applications in Algebra and Analysis at the Isaac Newton Institute for the Mathematical Sciences, whose hospitality is gratefully acknowledged.     

Koszulity for graded skew PBW extensions

Pre-Koszul and Koszul algebras were defined by Priddy [15 Priddy, S. (1970). Koszul resolutions. Trans. Am. Math. Soc. 152:39–60.[Crossref] , [Google Scholar]]. There exist some relations between these algebras and the skew PBW extensions defined in [8 Gallego, C., Lezama, O. (2011). Gröbner bases for ideals of σ-PBW extensions. Comm. Algebra 39(1):50–75.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. In [24 Suárez, H., Reyes, A. (submitted for publications). Koszulity for skew PBW extensions over fields. [Google Scholar]] we gave conditions to guarantee that skew PBW extensions over fields it turns out homogeneous pre-Koszul or Koszul algebra. In this paper we complement these results defining graded skew PBW extensions and showing that if R is a finite presented Koszul 𝕂-algebra then every graded skew PBW extension of R is Koszul.     

Armendariz modules over skew PBW extensions

The aim of this article is to develop the theory of skew Armendariz and quasi-Armendariz modules over skew PBW extensions. We generalize the results of several works in the literature concerning these topics for the context of Ore extensions to another non-commutative rings which can not be expressed as iterated Ore extensions. As a consequence of our treatment, we extend and unify different results about the Armendariz, Baer, p.p., and p.q.-Baer properties for Ore extensions and skew PBW extensions.     

Automorphisms of ore extensions by skew derivations

Let R be a prime ring and δ a σ-derivation of R, where σ is an automorphism of R. Our aim (Theorem 1) is to determine any automorphism S of R[t;σ,δ] satisfying S(A)?R for a one-sided ideal A≠0 of R by dropping the assumption t∈R[t;σ,δ]. As an application, if R is a domain, it is shown that any X-inner automorphism of R[t;σ,δ] stabilizes R.     

K-groups of quadratic extensions of rings

We construct the Levin braid connecting the Tate cohomology ofK-groups for quadratic extensions of rings with antistructures. Explicit formulas for isomorphisms of relative cohomology groups for the induced mapping and for the transfer mapping are obtained; these formulas are necessary in the construction of the Levin braid.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 272–280, August, 1995.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-011-1402.     

On free quadratic extensions of rings

The quaternion algebraB[j] over a commutative ringB with 1 defined byS. Parimala andR. Sridharan is generalized in two directions: (1) the ringB may be non-commutative with 1, and (2)j ² may be any invertible element (not necessarily –1). LetG={} be an automorphism group ofB of order 2, andA={b inB| (b)=b}. LetB[j] be a generalized quaternion algebra such thataj (a) for eacha inB. It will be shown thatB is Galois (for non-commutative ring extensions) overA which is contained in the center ofB if and only ifB[j] is Azumaya overA. Also,A[j] is a splitting ring forB[j] such thatA[j] is Galois overA. Moreover, we shall determine which automorphism group ofA[j] is a Galois group.     

Cohomology,extensions and automorphisms of skew braces

The second cohomology group of a left skew brace with coefficients in a trivial left brace with non-trivial actions is defined, its connection with extensions of a left skew brace by a trivial brace is established and a Wells' like exact sequence relating the second cohomology group with inducible automorphisms of an extension of left skew braces is constructed.