Azumaya Algebra | ScienceGate

Let R be a commutative ring, G a finite abelian group of order n and exponent m, and assume n is a unit in R. In [10], F. W. Long defined a generalized Brauer group, BD(R, G), of algebras with a G-action and G-grading, whose elements are equivalence classes of G-Azumaya algebras. In this paper we investigate the automorphisms of a G-Azumaya algebra A and prove that if Picm(R) is trivial, then these automorphisms are all, in some sense, inner.In fact, each of these “inner” automorphisms can be written as the composition of an inner automorphism in the usual sense and a “linear“ automorphism, i.e., an automorphism of the typewith r(σ) a unit in R. We then use these results to show that the group of gradings of the centre of a G-Azumaya algebra A is a direct summand of G, and thus if G is cyclic of order pr, A is the (smash) product of a commutative and a central G-Azumaya algebra.

Download Full-text

Totally Integrally Closed Azumaya Algebras

Canadian Mathematical Bulletin ◽

10.4153/cmb-1990-065-5 ◽

1990 ◽

Vol 33 (4) ◽

pp. 398-403

Author(s):

R. Macoosh ◽

R. Raphael

Keyword(s):

Semiprime Ring ◽

Commutative Rings ◽

Azumaya Algebras ◽

Azumaya Algebra ◽

Integral Extension ◽

Integrally Closed ◽

Carry Over

AbstractEnochs introduced and studied totally integrally closed rings in the class of commutative rings. This article studies the same question for Azumaya algebras, a study made possible by Atterton's notion of integral extensions for non-commutative rings.The main results are that Azumaya algebras are totally integrally closed precisely when their centres are, and that an Azumaya algebra over a commutative semiprime ring has a tight integral extension that is totally integrally closed. Atterton's integrality differs from that often studied but is very natural in the context of Azumaya algebras. Examples show that the results do not carry over to free normalizing or excellent extensions.

Download Full-text

The octonions form an Azumaya algebra in certain braided linear Gr-categories

Advances in Applied Clifford Algebras ◽

10.1007/s00006-016-0656-z ◽

2016 ◽

Vol 27 (2) ◽

pp. 1055-1064 ◽

Cited By ~ 1

Author(s):

Tao Cheng ◽

Hua-Lin Huang ◽

Yuping Yang ◽

Yinhuo Zhang

Keyword(s):

Azumaya Algebra

Download Full-text

On galois extensions of an azumaya algebra

Communications in Algebra ◽

10.1080/00927879708825959 ◽

1997 ◽

Vol 25 (6) ◽

pp. 1873-1882 ◽

Cited By ~ 6

Author(s):

Ricardo Alfaro ◽

George Szeto

Keyword(s):

Galois Extensions ◽

Azumaya Algebra

Download Full-text

Maximal orders in an azumaya algebra over a von neumann regular ring

Communications in Algebra ◽

10.1080/00927878308822848 ◽

1983 ◽

Vol 11 (3) ◽

pp. 257-286

Author(s):

Bo Stenström

Keyword(s):

Regular Ring ◽

Von Neumann Regular Ring ◽

Azumaya Algebra ◽

Von Neumann ◽

Maximal Orders ◽

Von Neumann Regular

Download Full-text

An Azumaya Algebra Version of the Kneser–Tits Problem for Groups of TypeD4

Communications in Algebra ◽

10.1080/00927870903447221 ◽

2010 ◽

Vol 39 (1) ◽

pp. 133-152

Author(s):

L. H. Rowen ◽

D. Saltman ◽

Y. Segev ◽

U. Vishne

Keyword(s):

Azumaya Algebra

Download Full-text

CENTRALIZERS AND INDUCTION

Journal of Algebra and Its Applications ◽

10.1142/s0219498807002363 ◽

2007 ◽

Vol 06 (03) ◽

pp. 505-526 ◽

Cited By ~ 1

Author(s):

LARS KADISON

Keyword(s):

Endomorphism Ring ◽

Ring Homomorphism ◽

Normal Subgroups ◽

Ring Extension ◽

Azumaya Algebra ◽

Equivalence Of Categories ◽

Tensor Square ◽

Special Case

Given a ring homomorphism B → A, consider its centralizer R = AB, bimodule endomorphism ring S = End BAB and sub-tensor-square ring T = (A ⊗ BA)B. Nonassociative tensoring by the cyclic modules RT or SR leads to an equivalence of categories inverse to the functors of induction of restricted A-modules or restricted coinduction of B-modules in case A | B is separable, H-separable, split or left depth two (D2). If RT or SR are projective, this property characterizes separability or splitness for a ring extension. Only in the case of H-separability is RT a progenerator, which takes the place of the key module AAe for an Azumaya algebra A. In addition, we characterize left D2 extensions in terms of the module TR, and show that the centralizer of a depth two extension is a normal subring in the sense of Rieffel as well as pre-braided commutative. For example, the notion of normality yields a version for Hopf subalgebras of the fact that normal subgroups have normal centralizers, and yields a special case of a conjecture that D2 Hopf subalgebras are normal.

Download Full-text