Finite Dimensional Central Simple Algebra Research Articles

-Derivations of group rings

We study -derivations of a group ring RG of a finite group G over an integral domain R with 1. As an application we extend a well known result on derivation of an integral group ring to -derivation on it for a finite group G with some conditions on σ and τ. In the process of the extension, a generalization of an application of Skolem-Noether Theorem to derivation on a finite dimensional central simple algebra has also been given for the -derivation case.

Symmetric Elements, Hermitian Forms, and Cyclic Involutions

Let k be a field, K/k be a quadratic separable field extension, and 𝒜 a finite dimensional central simple algebra over K. If k is global or the field of fractions of a two-dimensional excellent henselian local domain with an algebraically closed residue field of characteristic zero and the degree of 𝒜 is odd, we prove that all K/k-involutions on 𝒜 are cyclic.

Additive maps having a generalized derivation expansion

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.

Multiplication algebra and maps determined by zero products

Let A be a finite dimensional central simple algebra. By the Skolem–Noether theorem, every automorphism of A is inner. We will give a short proof of a somewhat more general result. The concept behind this proof is the fact that every linear map on A belongs to the multiplication algebra of A. As an application we will describe linear maps α, β : A → A such that α(x)β(y) = 0 whenever xy = 0.

Graded identities of matrix algebras and the universal graded algebra

In the last decade, group gradings and graded identities of finite dimensional central simple algebras have been an active area of research. We refer the reader to Bahturin, et al [6] and [7]. There are two basic kinds of group grading, elementary and fine. It was proved by Bahturin and Zaicev [7] that any group grading of Mn(C) is given by a certain composition of an elementary grading and a fine grading. In this paper we are concerned with fine gradings on Mn(C) and their corresponding graded identities. Let R be a simple algebra, finite dimensional over its center k and G a finite group. We say that R is fine graded by G if R ∼= ⊕g∈GRg is a grading and dimk(Rg) ≤ 1. Thus any component is either 0 or isomorphic to k as a k–vector space. It is easy to show that Supp(R), the subset of elements of G for which Rg is not 0, is a subgroup of G. Moreover

Tame group actions on central simple algebras

We study actions of linear algebraic groups on finite-dimensional central simple algebras. We describe the fixed algebra for a broad class of such actions.

A Note on Regular Crossed Products and Galois Representations

A crossed product representing an associative finite dimensional central simple algebra over a field is called regular if all values of the corresponding cocycle are roots of unity. Under a certain assumption such a crossed product is shown to allow the construction of Galois representations. The case of number fields is investigated more closely and several examples are discussed.

Higher trace forms and essential dimension in central simple algebras

We show that the essential dimension of a finite-dimensional central simple algebra coincides with the essential dimension of its r-linear trace form, $$ (a_1, \ldots, a_r) \mapsto tr(a_1, \ldots a_r), $$ for any r ≥ 3.

On bilinear maps on matrices with applications to commutativity preservers

Let M n be the algebra of all n × n matrices over a commutative unital ring C , and let L be a C -module. Various characterizations of bilinear maps { . , . } : M n × M n → L with the property that { x , y } = 0 whenever x any y commute are given. As the main application of this result we obtain the definitive solution of the problem of describing (not necessarily bijective) commutativity preserving linear maps from M n into M n for the case where C is an arbitrary field; moreover, this description is valid in every finite-dimensional central simple algebra.

Commutativity preserving linear maps on central simple algebras

Let F be a field with char ( F ) = 0 , and let A be a finite dimensional central simple algebra over F with dim F A ≠ 4 . If a linear map ϕ : A → A satisfies ϕ ( x 2 ) ϕ ( x ) = ϕ ( x ) ϕ ( x 2 ) for all x ∈ A (in particular, if ϕ preserves commutativity), then ϕ is either a standard commutativity preserving map or its range is commutative.

Group Actions on Central Simple Algebras

Let G be a group, F a field, and A a finite-dimensional central simple algebra over F on which G acts by F -algebra automorphisms. We study the subalgebras and ideals of A which are preserved by the group action. We prove a structure theorem and two classification theorems for invariant subalgebras under suitable hypotheses on A . We illustrate these results in the case of compact connected Lie groups and give some other applications. We also classify invariant ideals.

The minimal polynomial of a matrix*

Let Rbe a finite dimensional central simple algebra over a field F A be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial qA (λ) of A over F. By using qA (λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix.

Generators of Central Simple Algebras

LetAbe a finite-dimensional central simple algebra and letkbe a subfield of its centerZ(A). We say thatz1,…,zmgenerateAas a central simple algebra overkifA=S−1k[z1,…,zm], whereS=k[z1,…,zm]∩Z(A)*. In this paper we give a necessary and sufficient condition forAto be generated bymelements as a central simple algebra overk.

Involutions and Brauer-Severi varieties

For an arbitrary finite-dimensional central simple algebra A, we define central simple algebras s 2 A and λ 2 A which are Brauer-equivalent to A ⊗ A. Following an idea of Tamagawa, a correspondence between involutions on A and rational points on the Brauer-Severi varieties of s 2 A op and λ 2 A op is established in a characteristic-free context.

The solvability of linear matrix equations over a central simple algebra

Let R be a finite dimensional central simple algebra over a field F. By Using the method of representation, this paper discusses the linear matrix equations over R, obtains several necessary and sufficient conditions for existence of a solution or a unique solution, gives several explicit formulas of the unique solution for the equation AX − XB = C or X − AXB = C over R, and extends a Roth's theorem to the matrix equation over R.

A question on the discriminants of involutions of central division algebras

Let k be a field of characteristic not equal to 2 and D a finite dimensional central simple algebra over k whose class is 2-torsion in the Brauer group Br(k) of k. By classical results of Albert [A, Ch. X, Th. 19], D admits an involution which is identity on k. An invariant with values in k* modulo squares in k* called

Homogeneous Polynomials, Centralizers and Derivations in Rings

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.

Prime ideals in noncommutative valuation rings in finite dimensional central simple algebras

Prime ideals in noncommutative valuation rings in finite dimensional central simple algebras

A Noether Skolem theorem for group-graded rings

The classical Noether Skolem theorem gives conditions to determine if an automorphism or a derivation of a finite-dimensional central simple algebra is inner. The definition of inner has been extended to more general Hopf algebra actions. In this paper, we study group-gradings of central simple algebras and we give sufficient conditions for the grading to be inner. This provides a partial answer to a question of J. Bergen and S. Montgomery.

Generalized identities of finite dimensional central simple algebras

Generalized identities of finite dimensional central simple algebras