Intrinsic Functions on Matrices of Real Quaternions

1963 ◽  
Vol 15 ◽  
pp. 456-466 ◽  
Author(s):  
C. G. Cullen
Keyword(s):  
Division Algebra ◽  
Matrix Algebra ◽  
Simple Algebra ◽  
Real Field ◽  
Complex Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Simple Algebras ◽  
Real Quaternions ◽  
Algebra Over A Field

It is well known that any semi-simple algebra over the real field R, or over the complex field C, is a direct sum (unique except for order) of simple algebras, and that a finite-dimensional simple algebra over a field is a total matrix algebra over a division algebra, or equivalently, a direct product of a division algebra over and a total matrix algebra over (1). The only finite division algebras over R are R, C, and , the algebra of real quaternions, while the only finite division algebra over C is C.

Intrinsic Functions on Semi-Simple Algebras

1967 ◽  
Vol 19 ◽  
pp. 590-598 ◽  
Author(s):  
C. A. Hall
Keyword(s):  
Associative Algebra ◽  
Simple Algebra ◽  
Real Field ◽  
Complex Field ◽  
Ground Field ◽  
The Real ◽  
Simple Algebras

Rinehart (5) has introduced and motivated the study of the class of intrinsic functions on a linear associative algebra , with identity, over the real field R or the complex field C. In this paper we shall consider a semi-simple algebra = ⊕ … ⊕ over R or C with simple components . Let G be the group of all automorphisms or anti-automorphisms of which leave the ground field elementwise invariant, and let H be the subgroup of G such that Ω = (i = 1, 2, … , t) for each Ω in H.

Normed Right Alternative Algebras Over the Reals

1972 ◽  
Vol 24 (6) ◽  
pp. 1183-1186 ◽  
Author(s):  
José I. Nieto
Keyword(s):  
Division Algebra ◽  
Real Field ◽  
Division Algebras ◽  
Classical Theorem ◽  
Complex Field ◽  
The Real ◽  
Alternative Algebras ◽  
Finite Dimensional

One of the most interesting results on real normed division algebras says that every real normed associative division algebra is finite dimensional [6, Theorem 1.7.6], and hence by a classical theorem of Frobenius either isomorphic to the real field, the complex field, or the algebra of quaternions. Thus the dimension of the algebra can only be either 1, 2 or 4.

scholarly journals Admissible groups, symmetric factor sets, and simple algebras

1984 ◽  
Vol 7 (4) ◽  
pp. 707-711
Author(s):  
R. A. Mollin
Keyword(s):  
Division Algebra ◽  
Multiplicative Group ◽  
Sufficient Conditions ◽  
Simple Algebra ◽  
Roots Of Unity ◽  
Finite Dimensional ◽  
Odd Order ◽  
Simple Algebras ◽  
Necessary And Sufficient ◽  
Maximal Subfield

LetKbe a field of characteristic zero and suppose thatDis aK-division algebra; i.e. a finite dimensional division algebra overKwith centerK. In Mollin [1] we proved that ifKcontains no non-trivial odd order roots of unity, then every finite odd order subgroup ofD*the multiplicative group ofD, is cyclic. The first main result of this paper is to generalize (and simplify the proof of) the above. Next we generalize and investigate the concept of admissible groups. Finally we provide necessary and sufficient conditions for a simple algebra, with an abelian maximal subfield, to be isomorphic to a tensor product of cyclic algebras. The latter is achieved via symmetric factor sets.

Simple Algebras Over Rational Function Fields

1979 ◽  
Vol 31 (4) ◽  
pp. 831-835 ◽  
Author(s):  
T. Nyman ◽  
G. Whaples
Keyword(s):  
Rational Function ◽  
Number Field ◽  
Matrix Algebra ◽  
Function Fields ◽  
Simple Algebra ◽  
Noether Theorem ◽  
Full Matrix ◽  
Full Matrix Algebra ◽  
Rank One ◽  
Simple Algebras

The well-known Hasse-Brauer-Noether theorem states that a simple algebra with center a number field k splits over k (i.e., is a full matrix algebra) if and only if it splits over the completion of k at every rank one valuation of k. It is natural to ask whether this principle can be extended to a broader class of fields. In particular, we prove here the following extension.

Rings whose Indecomposable Injective Modules are Uniserial

1982 ◽  
Vol 34 (4) ◽  
pp. 797-805 ◽  
Author(s):  
David A. Hill
Keyword(s):  
Artinian Ring ◽  
Dimensional Algebra ◽  
Artinian Rings ◽  
Direct Sum ◽  
Injective Modules ◽  
Finite Dimensional ◽  
Uniserial Modules ◽  
Left And Right ◽  
Finitely Generated Modules ◽  
Algebra Over A Field

A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.

Isotopes of simple algebras of arbitrary dimension

2019 ◽  
Vol 13 (06) ◽  
pp. 2050108 ◽  
Author(s):  
Vita Glizburg ◽  
Sergey Pchelintsev
Keyword(s):  
Jordan Algebra ◽  
Matrix Algebra ◽  
Dimensional Space ◽  
Arbitrary Dimension ◽  
Simple Algebra ◽  
Symmetric Bilinear Form ◽  
Finite Dimensional Algebra ◽  
Ground Field ◽  
Dimensional Algebra ◽  
Finite Dimensional

It is proved that every finite-dimensional algebra is embeddable in a simple finite-dimensional algebra (a suitable isotope of a matrix algebra). An isotope of the 2nd order matrix algebra over an infinite extension of the ground field may contain a trivial ideal. Every one-sided isotope of a simple unital alternative or Jordan algebra is a simple algebra. Besides, any isotope of a central simple non-Lie Maltsev algebra of characteristic other than 2 and 3 is a simple algebra. But an isotope of a simple Jordan algebra of the symmetric bilinear form on the infinite dimensional space may contain a trivial ideal.

scholarly journals On Modular Representation Algebras and a Class of Matrix Algebras

1982 ◽  
Vol 33 (3) ◽  
pp. 351-355 ◽  
Author(s):  
J-C. Renaud
Keyword(s):  
Tensor Product ◽  
Cyclic Group ◽  
Prime Order ◽  
Matrix Algebra ◽  
Modular Representation ◽  
Complex Field ◽  
Matrix Algebras ◽  
Direct Sum ◽  
Standard Operations

AbstractLet G be a cyclic group of prime order p and K a field of characteristic p. The set of classes of isomorphic indecomposable (K, G)-modules forms a basis over the complex field for an algebra p (Green, 1962) with addition and multiplication being derived from direct sum and tensor product operations.Algebras n with similar properties can be defined for all n ≥ 2. Each such algebra is isomorphic to a matrix algebra Mn of n × n matrices with complex entries and standard operations. The characters of elements of n are the eigenvalues of the corresponding matrices in Mn.

Classes of Functions on Algebras

1966 ◽  
Vol 18 ◽  
pp. 139-146 ◽  
Author(s):  
C. G. Cullen ◽  
C. A. Hall
Keyword(s):  
Associative Algebra ◽  
Real Field ◽  
Complex Field ◽  
The Real ◽  
Finite Dimensional

Let be a finite-dimensional linear associative algebra over the real field R or the complex field C and let F be a function with domain and range in .Several classes of functions on have been discussed in the literature, and it is the purpose of this paper to discuss the relationships between these classes and to present some interesting examples. First we shall list the definitions of the classes we wish to consider here.

scholarly journals The Norm of a Skew Polynomial

2021 ◽  
Author(s):  
S. Pumplün ◽  
D. Thompson
Keyword(s):  
Division Algebra ◽  
Polynomial Ring ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Skew Polynomial Ring ◽  
Skew Polynomials ◽  
Finite Dimensional ◽  
Central Simple ◽  
Skew Polynomial ◽  
Reduced Norm

AbstractLet D be a finite-dimensional division algebra over its center and R = D[t;σ,δ] a skew polynomial ring. Under certain assumptions on δ and σ, the ring of central quotients D(t;σ,δ) = {f/g|f ∈ D[t;σ,δ],g ∈ C(D[t;σ,δ])} of D[t;σ,δ] is a central simple algebra with reduced norm N. We calculate the norm N(f) for some skew polynomials f ∈ R and investigate when and how the reducibility of N(f) reflects the reducibility of f.

UNIT GROUPS OF CENTRAL SIMPLE ALGEBRAS AND THEIR FRATTINI SUBGROUPS

2010 ◽  
Vol 09 (06) ◽  
pp. 921-932 ◽  
Author(s):  
R. FALLAH-MOGHADDAM ◽  
M. MAHDAVI-HEZAVEHI
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Global Field ◽  
Central Simple Algebras ◽  
Frattini Subgroup ◽  
Division Rings ◽  
Finite Dimensional ◽  
Simple Algebras ◽  
Central Simple

Given a finite dimensional F-central simple algebra A = Mn(D), the connection between the Frattini subgroup Φ(A*) and Φ(F*) via Z(A'), the center of the derived group of A*, is investigated. Setting G = F* ∩ Φ(A*), it is shown that [Formula: see text] where the intersection is taken over primes p not dividing the degree of A. Furthermore, when F is a local or global field, the group G is completely determined. Using the above connection, Φ(A*) is also calculated for some particular division rings D.

Sign in / Sign up

Export Citation Format

Share Document