The Brauer Group of a Commutative Ring

1993 ◽  
pp. 185-198
Author(s):  
Benson Farb ◽  
R. Keith Dennis
Keyword(s):  
Commutative Ring ◽  
Brauer Group

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 The Brauer group of a commutative ring

1960 ◽  
Vol 97 (3) ◽  
pp. 367-367 ◽  
Author(s):  
Maurice Auslander ◽  
Oscar Goldman
Keyword(s):  
Commutative Ring ◽  
Brauer Group

Correction to on the Brauer Group of Algebras Having a Grading and an Action

1980 ◽  
Vol 32 (6) ◽  
pp. 1523-1524 ◽  
Author(s):  
Morris Orzech
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Finite Type ◽  
Maximal Ideal ◽  
Finite Abelian Group ◽  
Brauer Group ◽  
Erroneous Result

Let R be a commutative ring, G a finite abelian group. Let A be an R-algebra which is graded by G (i.e. A = Σ⊕σ∈GAσ, where AσAτ ⊂ Aστ for σ, τ in G) and for which A1 is an R-module of finite type. In Remark 4.1 (a) of [1] we asserted that under these hypotheses if u is in A and u + pA is homogeneous in A/pA for each maximal ideal p of R then u is homogeneous in A. We used this assertion for u a unit in A such that a → uau–1 is a grading-preserving homomorphism. K. Ulbrich has kindly pointed out a counterexample to the assertion: R = Z/4Z, G = {1, σ};, u = 2σ + 1, p = 2R. Proposition 4.2 of [1] uses the erroneous result and is in turn invoked later in the paper.

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.

Embedding the Hopf Automorphism Group into the Brauer Group

1998 ◽  
Vol 41 (3) ◽  
pp. 359-367 ◽  
Author(s):  
Fred Van Oystaeyen ◽  
Yinhuo Zhang
Keyword(s):  
Automorphism Group ◽  
Exact Sequence ◽  
Hopf Algebra ◽  
Commutative Ring ◽  
Brauer Group

AbstractLet H be a faithfully projective Hopf algebra over a commutative ring k. In [8, 9] we defined the Brauer group BQ(k, H) of H and an homomorphism π from Hopf automorphism group AutHopf(H) to BQ(k,H). In this paper, we show that the morphism π can be embedded into an exact sequence.

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.

Brauer–Clifford Group of Lie–Rinehart Algebras

2022 ◽  
Vol 29 (01) ◽  
pp. 99-112
Author(s):  
Thomas Guédénon
Keyword(s):  
Differential Geometry ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Commutative Algebra ◽  
Monoidal Category ◽  
Brauer Group ◽  
Azumaya Algebras ◽  
Symmetric Monoidal Category ◽  
Clifford Group

In this paper we define the notion of Brauer–Clifford group for [Formula: see text]-Azumaya algebras when [Formula: see text] is a commutative algebra and[Formula: see text] is a [Formula: see text]-Lie algebra over a commutative ring [Formula: see text]. This is the situation that arises in applications having connections to differential geometry. This Brauer–Clifford group turns out to be an example of a Brauer group of a symmetric monoidal category.

On quasi-radical of near-ring of polynomials

2019 ◽  
Vol 56 (2) ◽  
pp. 252-259
Author(s):  
Ebrahim Hashemi ◽  
Fatemeh Shokuhifar ◽  
Abdollah Alhevaz
Keyword(s):  
Commutative Ring ◽  
Nilpotent Elements ◽  
Ring Of Polynomials

Abstract The intersection of all maximal right ideals of a near-ring N is called the quasi-radical of N. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials R0[x] equals to the set of all nilpotent elements of R0[x], when R is a commutative ring with Nil (R)2 = 0. Then we show that the quasi-radical of R0[x] is a subset of the intersection of all maximal left ideals of R0[x]. Also, we give an example to show that for some commutative ring R the quasi-radical of R0[x] coincides with the intersection of all maximal left ideals of R0[x]. Moreover, we prove that the quasi-radical of R0[x] is the greatest quasi-regular (right) ideal of it.

N-ideals of commutative rings

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2933-2941 ◽  
Author(s):  
Unsal Tekir ◽  
Suat Koc ◽  
Kursat Oral
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Nonzero Identity

In this paper, we present a new classes of ideals: called n-ideal. Let R be a commutative ring with nonzero identity. We define a proper ideal I of R as an n-ideal if whenever ab ? I with a ? ?0, then b ? I for every a,b ? R. We investigate some properties of n-ideals analogous with prime ideals. Also, we give many examples with regard to n-ideals.

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