A characterization of Azumaya algebras

AbstractLet R be a commutative ring with identity 1, and A a finitely generated R-algebra. It is shown that A is an Azumaya R-algebra if and only if every stalk of the Pierce sheaf induced by A is an Azumaya algebra

On Galois projective group rings

Let A be a ring with 1, C the center of A and G′ an inner automorphism group of A induced by {Uα in ​A/α in a finite group G whose order is invertible}. Let AG′ be the fixed subring of A under the action of G′.If A is a Galcis extension of AG′ with Galois group G′ and C is the center of the subring ∑αAG′Uα then A=∑αAG′Uα and the center of AG′ is also C. Moreover, if ∑αAG′Uα is Azumaya over C, then A is a projective group ring

The Galois extensions induced by idempotents in a Galois algebra

Let B be a Galois algebra with Galois group G, Jg={b∈B|bx=g(x)b   for all   x∈B} for each g∈G, eg the central idempotent such that BJg=Beg, and eK=∑g∈K,eg≠1eg for a subgroup K of G. Then BeK is a Galois extension with the Galois group G(eK)(={g∈G|g(eK)=eK}) containing K and the normalizer N(K) of K in G. An equivalence condition is also given for G(eK)=N(K), and BeG is shown to be a direct sum of all Bei generated by a minimal idempotent ei. Moreover, a characterization for a Galois extension B is shown in terms of the Galois extension BeG and B(1−eG)

On central commutator Galois extensions of rings

Let B be a ring with 1, G a finite automorphism group of B of order n for some integer n, BG the set of elements in B fixed under each element in G, and Δ=VB(BG) the commutator subring of BG in B. Then the type of central commutator Galois extensions is studied. This type includes the types of Azumaya Galois extensions and Galois H-separable extensions. Several characterizations of a central commutator Galois extension are given. Moreover, it is shown that when G is inner, B is a central commutator Galois extension of BG if and only if B is an H-separable projective group ring BGGf. This generalizes the structure theorem for central Galois algebras with an inner Galois group proved by DeMeyer

The general Ikehata theorem for H

Let B be a ring with 1,   C the center of B,   G an automorphism group of B of order n for some integer n,   CG the set of elements in C fixed under G,   Δ=Δ(B,G,f) a crossed product over B where f is a factor set from G×G to U(CG). It is shown that Δ is an H-separable extension of B and VΔ(B) is a commutative subring of Δ if and only if C is a Galois algebra over CG with Galois group G|C≅G

On generalized quaternion algebras

Let B be a commutative ring with 1, and G(={σ}) an automorphism group of B of order 2. The generalized quaternion ring extension B[j] over B is defined by S. Parimala and R. Sridharan such that (1) B[j] is a free B-module with a basis {1,j}, and (2) j2=−1 and jb=σ(b)j for each b in B. The purpose of this paper is to study the separability of B[j]. The separable extension of B[j] over B is characterized in terms of the trace (=1+σ) of B over the subring of fixed elements under σ. Also, the characterization of a Galois extension of a commutative ring given by Parimala and Sridharan is improved

On free ring extensions of degree n

Nagahara and Kishimoto [1] studied free ring extensions B(x) of degree n for some integer n over a ring B with 1, where xn=b, cx=xρ(c) for all c and some b in B(ρ=automophism of  B), and {1,x…,xn−1} is a basis. Parimala and Sridharan [2], and the author investigated a class of free ring extensions called generalized quaternion algebras in which b=−1 and ρ is of order 2. The purpose of the present paper is to generalize a characterization of a generalized quaternion algebra to a free ring extension of degree n in terms of the Azumaya algebra. Also, it is shown that a one-to-one correspondence between the set of invariant ideals of B under ρ and the set of ideals of B(x) leads to a relation of the Galois extension B over an invariant subring under ρ to the center of B

On separable abelian extensions of rings

Let R be a ring with 1, G(=〈ρ1〉×…×〈ρm〉) a finite abelian automorphism group of R of order n where 〈ρi〉 is cyclic of order ni. for some integers n, ni, and m, and C the center of R whose automorphism group induced by G is isomorphic with G. Then an abelian extension R[x1,…,xm] is defined as a generalization of cyclic extensions of rings, and R[x1,…,xm] is an Azumaya algebra over K(=CG={c   in   C/(c)ρi=c   for each   ρi   in   G}) such that R[x1,…,xm]≅RG⊗KC[x1,…,xm] if and only if C is Galois over K with Galois group G (the Kanzaki hypothesis)

Notes on Galois algebras

Let B be a ring with 1, C the center of B, and G an automorphism group of B of order n for some integer n. Assume B is a Galois algebra over R with Galois group G. For a nonzero idempotent e R, it the rank of Be over Ce is defined and equal to the order of H|Be where H = {g G | g(c) = c for each c C}, then Be is a central Galois algebra with Galois group H|Be. This generalizes the F. R. DeMeyer and T. Kanzaki theorems for Galois algebras. Moreover, a structure theorem for a Galois algebra is given in terms of the concept of the rank of projective module