scholarly journals More on the Schur group of a commutative ring

1985 ◽  
Vol 8 (2) ◽  
pp. 275-282 ◽  
Author(s):  
R. A. Mollin
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Homomorphic Image ◽  
Group Ring ◽  
Local Field ◽  
Natural Generalization ◽  
Brauer Group ◽  
Central Simple Algebras ◽  
Simple Algebras ◽  
Central Simple

The Schur group of a commutative ring,R, with identity consists of all classes in the Brauer group ofRwhich contain a homomorphic image of a group ringRGfor some finite groupG. It is the purpose of this article to continue an investigation of this group which was introduced in earlier work as a natural generalization of the Schur group of a field. We generalize certain facts pertaining to the latter, among which are results on extensions of automorphisms and decomposition of central simple algebras into a product of cyclics. Finally we introduce the Schur exponent of a ring which equals the well-known Schur index in the global or local field case.

scholarly journals More on the Schur group of a commutative ring

1985 ◽  
Vol 8 (3) ◽  
pp. 513-520
Author(s):  
R. A. Mollin
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Homomorphic Image ◽  
Group Ring ◽  
Local Field ◽  
Natural Generalization ◽  
Brauer Group ◽  
Central Simple Algebras ◽  
Simple Algebras ◽  
Central Simple

The Schur group of a commutative ring,R, with identity consists of all classes in the Brauer group ofRwhich contain a homomorphic image of a group ringRGfor some finite groupG. It is the purpose of this article to continue an investigation of this group which was introduced in earler work as a natural generalization of the Schur group of a field. We generalize certain facts pertaining to the latter, among which are results on extensions of automorphisms and decomposition of central simple algebras into a product of cyclics. Finally we introduce the Schur exponent of a ring which equals the well-known Schur index in the global or local field case.

On the Brauer Group of Algebras Having a Grading and an Action

1976 ◽  
Vol 28 (3) ◽  
pp. 533-552 ◽  
Author(s):  
Morris Orzech
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Bilinear Form ◽  
Brauer Group ◽  
Central Simple Algebras ◽  
Simple Algebras ◽  
Central Simple

Beginning with Wall's introduction [19] of Z2-graded central simple algebras over a field K, a number of related generalizations of the Brauer group have been proposed. In [16] the field K was replaced by a commutative ring R, building upon the theory developed in [1]. The concept of a G-graded central simple K-algebra (G an abelian group) was first defined in [12]; this work and that of [16] was subsequently unified in [6] and [7] via the construction and computation of the graded Brauer group Bφ﹛R, G) (φ a bilinear form from G X G to U(R), the units of R).

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.

scholarly journals Derivations in central separable algebras

1978 ◽  
Vol 19 (1) ◽  
pp. 75-77 ◽  
Author(s):  
George T. Georgantas
Keyword(s):  
Galois Group ◽  
Brauer Group ◽  
Normal Extension ◽  
Noether Theorem ◽  
Central Simple Algebras ◽  
Group Extensions ◽  
Simple Algebras ◽  
Central Simple ◽  
Inner Automorphisms ◽  
Separable Algebras

Given N a finite separable normal extension of a field F, it is well known that the Brauer group Br(N/F) of classes of central simple F-algebras split by N is isomorphic with Ext(N*, G), the classes of group extensions of N* by the Galois group G of N over F. In the construction of this isomorphism, a key role is played by the Skolem-Noether Theorem which extends automorphisms to inner automorphisms in central simple algebras.

scholarly journals Higher Derivations and Central Simple Algebras

1968 ◽  
Vol 32 ◽  
pp. 21-30 ◽  
Author(s):  
A. Roy ◽  
R. Sridharan
Keyword(s):  
Commutative Ring ◽  
Equivalence Classes ◽  
Central Simple Algebras ◽  
Isomorphism Classes ◽  
Simple Algebras ◽  
Central Simple

Let K be a commutative ring, A a K-algebra, and B a K-subalgebra of A. The object of this paper is to prove some results on higher derivations (in the sense of Jacobson [4]) of B into A. In § 1 we introduce a notion of equivalence among higher derivations. With this notion of equivalence, we prove in § 2 (Theorem 1) that the equivalence classes of higher K-derivations of B into A are in one-one correspondence with the isomorphism classes of certain filtered B ⊗ KA°-modules, where A° denotes the opposite algebra of A.

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