Finite dimensional algebras without nilpotents over algebraically closed fields

1979 ◽  
Vol 32 (1) ◽  
pp. 10-12 ◽  
Author(s):  
Helmut R�hrl
Keyword(s):  
Finite Dimensional ◽  
Closed Fields ◽  
Algebraically Closed Fields ◽  
Finite Dimensional Algebras

scholarly journals Maximal subalgebras of finite-dimensional algebras

2019 ◽  
Vol 31 (5) ◽  
pp. 1283-1304 ◽  
Author(s):  
Miodrag Cristian Iovanov ◽  
Alexander Harris Sistko
Keyword(s):  
Associative Algebra ◽  
Complete Classification ◽  
Maximal Subalgebras ◽  
Incidence Algebras ◽  
Finite Dimensional ◽  
Semisimple Algebras ◽  
Closed Fields ◽  
Representations Of Algebras ◽  
Algebraically Closed Fields ◽  
Full Classification

AbstractWe study maximal associative subalgebras of an arbitrary finite-dimensional associative algebra B over a field {\mathbb{K}} and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case and then lifting to non-semisimple algebras. The results are sharpest in the case of algebraically closed fields and take special forms for algebras presented by quivers with relations. We also relate representation theoretic properties of the algebra and its maximal and other subalgebras and provide a series of embeddings between quivers, incidence algebras and other structures which relate indecomposable representations of algebras and some subalgebras via induction/restriction functors. Some results in literature are also re-derived as a particular case, and other applications are given.

scholarly journals Involutions on finite-dimensional algebras over real closed fields

2004 ◽  
Vol 77 (1) ◽  
pp. 123-128 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Characteristic Zero ◽  
Real Closed Field ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Real Closed Fields ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

scholarly journals Representation-Directed Diamonds

2001 ◽  
Vol 4 ◽  
pp. 14-21
Author(s):  
Peter Dräxler
Keyword(s):  
Finite Dimensional Algebra ◽  
Program System ◽  
Dimensional Algebra ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Positive Roots ◽  
Algebraically Closed Fields

AbstractA module over a finite-dimensional algebra is called a ‘diamond’ if it has a simple top and a simple socle. Using covering theory, the classification of all diamonds for algebras of finite representation type over algebraically closed fields can be reduced to representation-directed algebras. The author proves a criterion referring to the positive roots of the corresponding Tits quadratic form, which makes it easy to check whether a representation-directed algebra has a faithful diamond. Using an implementation of this criterion in the CREP program system on representation theory, he is able to classify all exceptional representation-directed algebras having a faithful diamond. He obtains a list of 157 algebras up to isomorphism and duality. The 52 maximal members of this list are presented at the end of this paper.

Presentations of skew fields. I. Existentially closed skew fields and the Nullstellensatz

1975 ◽  
Vol 77 (1) ◽  
pp. 7-19 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Algebraic Geometry ◽  
Matrix Ring ◽  
Commutative Rings ◽  
Commutative Case ◽  
Matrix Rings ◽  
Finite Dimensional ◽  
Existentially Closed ◽  
Closed Fields ◽  
Skew Fields ◽  
Algebraically Closed Fields

1. Introduction. The Nullstellensatz in commutative algebraic geometry may be described as a means of studying certain commutative rings (viz. affine algebras) by their homomorphisms into algebraically closed fields, and a number of attempts have been made to extend the result to the non-commutative case. In particular, Amitsur and Procesi have studied the case of general rings, with homomorphisms into matrix rings over commutative fields ((1), (2)) and Procesi has obtained more precise results for homomorphisms of PI-rings (11). Since a finite-dimensional division algebra can always be embedded in a matrix ring over a field, this includes the case of skew fields that are finite-dimensional over their centre, but it tells us nothing about general skew fields.

Preservation of elementary equivalence under scalar extension

1982 ◽  
Vol 47 (4) ◽  
pp. 734-738
Author(s):  
Bruce I. Rose
Keyword(s):  
Identity Element ◽  
Positive Answer ◽  
General Question ◽  
Division Algebras ◽  
Extension Field ◽  
Elementary Equivalence ◽  
Applied Model ◽  
Finite Dimensional ◽  
Positive Statements ◽  
Finite Dimensional Algebras

In this note we show that taking a scalar extension of two elementarily equivalent finite-dimensional algebras over the same field preserves elementary equivalence. The general question of whether or not tensor product preserves elementary equivalence was originally raised in [4]. In [3] Feferman relates an example of Ersov which answers the question negatively. Eklof and Olin [7] also provide a counterexample to the general question in the context of two-sorted structures. Thus the result proved below is a partial positive answer to a general question whose status has been resolved negatively. From the viewpoint of applied model theory it seems desirable to find contexts in which positive statements of preservation can be obtained. Our result does have an application; a corollary to it increases our understanding of what it means for two division algebras to be elementarily equivalent.All algebras are finite-dimensional algebras over fields. All algebras contain an identity element, but are not necessarily associative.Recall that the center of a not necessarily associative algebra A is the set of elements which commute and “associate” with all elements of A. The notion of a scalar extension is an important one in algebra. If A is an algebra over F and G is an extension field of F, then the scalar extension of A by G is the algebra A ⊗F G.

scholarly journals Perfect pseudo-algebraically closed fields are algebraically bounded

2004 ◽  
Vol 271 (2) ◽  
pp. 627-637 ◽  
Author(s):  
Zoé Chatzidakis ◽  
Ehud Hrushovski
Keyword(s):  
Closed Fields ◽  
Algebraically Closed Fields
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