MAXIMAL ORDERS IN A FINITE-DIMENSIONAL CENTRAL SIMPLE ALGEBRA OVER A VALUATION RING OF HEIGHT 1

1980 ◽  
Vol 36 (4) ◽  
pp. 483-493 ◽  
Author(s):  
N I Dubrovin
Keyword(s):  
Valuation Ring ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Maximal Orders ◽  
Finite Dimensional ◽  
Central Simple

Additive maps having a generalized derivation expansion

2015 ◽  
Vol 14 (04) ◽  
pp. 1550048 ◽  
Author(s):  
Tsiu-Kwen Lee
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Generalized Derivation ◽  
Elementary Operator ◽  
Additive Map ◽  
Finite Dimensional ◽  
Left And Right ◽  
Central Simple ◽  
Skolem Theorem ◽  
Additive Maps

Let R be a prime ring with extended centroid C. We prove that an additive map from R into RC + C can be characterized in terms of left and right b-generalized derivations if it has a generalized derivation expansion. As a consequence, a generalization of the Noether–Skolem theorem is proved among other things: A linear map from a finite-dimensional central simple algebra into itself is an elementary operator if it has a generalized derivation expansion.

scholarly journals Local selectivity of orders in central simple algebras

2017 ◽  
Vol 13 (04) ◽  
pp. 853-884 ◽  
Author(s):  
Benjamin Linowitz ◽  
Thomas R. Shemanske
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Ring Of Integers ◽  
Isomorphism Classes ◽  
Maximal Orders ◽  
Local Variant ◽  
Simple Algebras ◽  
Central Simple ◽  
Global Principle

Let [Formula: see text] be a central simple algebra of degree [Formula: see text] over a number field [Formula: see text], and [Formula: see text] be a strictly maximal subfield. We say that the ring of integers [Formula: see text] is selective if there exists an isomorphism class of maximal orders in [Formula: see text] no element of which contains [Formula: see text]. In the present work, we consider a local variant of the selectivity problem and applications. We first prove a theorem characterizing which maximal orders in a local central simple algebra contain the global ring of integers [Formula: see text] by leveraging the theory of affine buildings for [Formula: see text] where [Formula: see text] is a local central division algebra. Then as an application, we use the local result and a local–global principle to show how to compute a set of representatives of the isomorphism classes of maximal orders in [Formula: see text], and distinguish those which are guaranteed to contain [Formula: see text]. Having such a set of representatives allows both algebraic and geometric applications. As an algebraic application, we recover a global selectivity result, and give examples which clarify the interesting role of partial ramification in the algebra.

scholarly journals The Norm of a Skew Polynomial

2021 ◽  
Author(s):  
S. Pumplün ◽  
D. Thompson
Keyword(s):  
Division Algebra ◽  
Polynomial Ring ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Skew Polynomial Ring ◽  
Skew Polynomials ◽  
Finite Dimensional ◽  
Central Simple ◽  
Skew Polynomial ◽  
Reduced Norm

AbstractLet D be a finite-dimensional division algebra over its center and R = D[t;σ,δ] a skew polynomial ring. Under certain assumptions on δ and σ, the ring of central quotients D(t;σ,δ) = {f/g|f ∈ D[t;σ,δ],g ∈ C(D[t;σ,δ])} of D[t;σ,δ] is a central simple algebra with reduced norm N. We calculate the norm N(f) for some skew polynomials f ∈ R and investigate when and how the reducibility of N(f) reflects the reducibility of f.

UNIT GROUPS OF CENTRAL SIMPLE ALGEBRAS AND THEIR FRATTINI SUBGROUPS

2010 ◽  
Vol 09 (06) ◽  
pp. 921-932 ◽  
Author(s):  
R. FALLAH-MOGHADDAM ◽  
M. MAHDAVI-HEZAVEHI
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Global Field ◽  
Central Simple Algebras ◽  
Frattini Subgroup ◽  
Division Rings ◽  
Finite Dimensional ◽  
Simple Algebras ◽  
Central Simple

Given a finite dimensional F-central simple algebra A = Mn(D), the connection between the Frattini subgroup Φ(A*) and Φ(F*) via Z(A'), the center of the derived group of A*, is investigated. Setting G = F* ∩ Φ(A*), it is shown that [Formula: see text] where the intersection is taken over primes p not dividing the degree of A. Furthermore, when F is a local or global field, the group G is completely determined. Using the above connection, Φ(A*) is also calculated for some particular division rings D.

Homogeneous Polynomials, Centralizers and Derivations in Rings

1993 ◽  
Vol 45 (1) ◽  
pp. 22-32
Author(s):  
Onofrio Mario Divincenzo ◽  
Rosa Sagona
Keyword(s):  
Homogeneous Polynomial ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Homogeneous Polynomials ◽  
Finite Dimensional ◽  
Central Simple ◽  
Power Central ◽  
Primitive Ring

AbstractLetdbe a non-zero derivation on a primitive ringRandƒ(x1,…,xn) a homogeneous polynomial of degreem. We prove that the conditiond(ƒ(r1,…,rn)t) = 0, for allr1,…,rn∈R, withtdepending onr1,…,rn, forcesRto be a finite dimensional central simple algebra andƒpower-central valued onR. We also obtain bounds on [R:Z(R)] in terms ofm.

scholarly journals Crossed Products and Maximal Orders

1965 ◽  
Vol 25 ◽  
pp. 165-174 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Sufficient Conditions ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Crossed Product ◽  
Quotient Field ◽  
Maximal Orders ◽  
Rank One ◽  
Central Simple ◽  
Necessary And Sufficient ◽  
Algebra Over A Field

Let I’ be a maximal order over a complete discrete rank one valuation ring R in a central simple algebra over the quotient field of R. The purpose of this paper is to determine necessary and sufficient conditions for I’ to be equivalent to a crossed product over a tamely ramified extension of R.It is a classical result that every central simple algebra over a field k is equivalent to a crossed product over a Galois extension of k. Furthermore, it has been proved by Auslander and Goldman in [2] that every central separable algebra over a local ring is equivalent to a crossed product over an unramified extension.

scholarly journals Motivic decomposition of compactifications of certain group varieties

2018 ◽  
Vol 2018 (745) ◽  
pp. 41-58
Author(s):  
Nikita A. Karpenko ◽  
Alexander S. Merkurjev
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Chow Ring ◽  
Prime Degree ◽  
Central Simple

Abstract Let D be a central simple algebra of prime degree over a field and let E be an {\operatorname{\mathbf{SL}}_{1}(D)} -torsor. We determine the complete motivic decomposition of certain compactifications of E. We also compute the Chow ring of E.

Isotropy of orthogonal involutions in characteristic two

2018 ◽  
Vol 17 (12) ◽  
pp. 1850240 ◽  
Author(s):  
A.-H. Nokhodkar
Keyword(s):  
Quadratic Form ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Characteristic Two ◽  
Central Simple ◽  
The Given

A totally singular quadratic form is associated to any central simple algebra with orthogonal involution in characteristic two. It is shown that the given involution is isotropic if and only if its corresponding quadratic form is isotropic.

scholarly journals Splitting of Algebras by Function Fields of One Variable

1966 ◽  
Vol 27 (2) ◽  
pp. 625-642 ◽  
Author(s):  
Peter Roquette
Keyword(s):  
Tensor Product ◽  
Function Fields ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Field Extension ◽  
Brauer Group ◽  
Central Simple Algebras ◽  
Natural Mapping ◽  
Simple Algebras ◽  
Central Simple

Let K be a field and (K) the Brauer group of K. It consists of the similarity classes of finite central simple algebras over K. For any field extension F/K there is a natural mapping (K) → (F) which is obtained by assigning to each central simple algebra A/K the tensor product which is a central simple algebra over F. The kernel of this map is the relative Brauer group (F/K), consisting of those A ∈(K) which are split by F.

The Coniveau Filtration on for Some Severi–Brauer Varieties

2018 ◽  
Vol 62 (3) ◽  
pp. 565-576
Author(s):  
Eoin Mackall
Keyword(s):  
Spectral Sequence ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Norm Group ◽  
Central Simple ◽  
Reduced Norm ◽  
Coniveau Filtration

AbstractWe produce an isomorphism $E_{\infty }^{m,-m-1}\cong \text{Nrd}_{1}(A^{\otimes m})$ between terms of the $\text{K}$-theory coniveau spectral sequence of a Severi–Brauer variety $X$ associated with a central simple algebra $A$ and a reduced norm group, assuming $A$ has equal index and exponent over all finite extensions of its center and that $\text{SK}_{1}(A^{\otimes i})=1$ for all $i>0$.

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