scholarly journals On some infinitely presented associative algebras

1973 ◽  
Vol 16 (3) ◽  
pp. 290-293 ◽  
Author(s):  
Jacques Lewin
Keyword(s):  
Associative Algebra ◽  
Commutative Algebra ◽  
Polynomial Identity ◽  
Finite Dimension ◽  
Negative Answer ◽  
Free Associative Algebra ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Commutator Ideal ◽  
Finitely Presented

We prove here that if F is a finitely generated free associative algebra over the field and R is an ideal of F, then F/R2 is finitely presented if and only if F/R has finite dimension. Amitsur, [1, p. 136] asked whether a finitely generated algebra which is embeddable in matrices over a commutative f algebra is necessarily finitely presented. Let R = F′, the commutator ideal of F, then [4, theorem 6], F/F′2 is embeddable and thus provides a negative answer to his question. Another such example can be found in Small [6]. We also show that there are uncountably many two generator I algebras which satisfy a polynomial identity yet are not embeddable in any algebra of n xn matrices over a commutative algebra.

Standard Gröbner-Shirshov Bases of Free Algebras Over Rings, I

1998 ◽  
Vol 08 (06) ◽  
pp. 689-726 ◽  
Author(s):  
Alexander A. Mikhalev ◽  
Andrej A. Zolotykh
Keyword(s):  
Associative Algebra ◽  
Standard Basis ◽  
Free Associative Algebra ◽  
Semigroup Algebras ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Standard Bases ◽  
Equality Problem ◽  
Algebra Over A Field ◽  
The Ideal

We consider standard bases of ideals of free associative algebras over rings. The main result of the article is a criterion for a subset of a free associative algebra to be a standard basis of the ideal it generates. Based on this result, we present an infinite algorithm to construct the reduced standard basis of an ideal. A generalization in case of some semigroup algebras is presented. We also describe a way to construct weak standard bases and reduced standard bases of ideals of a free associative algebra over an arbitrary finitely generated ring (over a finitely generated algebra over a field). Some examples of constructions of standard bases and of solutions of the equality problem are included.

COMPUTING NILPOTENT QUOTIENTS OF ASSOCIATIVE ALGEBRAS AND ALGEBRAS SATISFYING A POLYNOMIAL IDENTITY

2011 ◽  
Vol 21 (08) ◽  
pp. 1339-1355 ◽  
Author(s):  
BETTINA EICK
Keyword(s):  
Associative Algebra ◽  
Polynomial Identity ◽  
Maximal Class ◽  
Arbitrary Field ◽  
Effective Algorithm ◽  
Associative Algebras ◽  
Class C ◽  
Finitely Presented

We describe an effective algorithm to determine the maximal class-c quotient (or the maximal commutative class-c quotient) of a finitely presented associative algebra over an arbitrary field. As application, we investigate the relatively free d-generator algebras satisfying the identity xn = 0. In particular, we consider the identity x3 = 0 and the cases (d, n) = (2, 4) and (d, n) = (2, 5).

scholarly journals AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE ASSOCIATIVE ALGEBRA

2007 ◽  
Vol 17 (05n06) ◽  
pp. 923-939 ◽  
Author(s):  
A. BELOV-KANEL ◽  
A. BERZINS ◽  
R. LIPYANSKI
Keyword(s):  
Automorphism Group ◽  
Associative Algebra ◽  
Free Associative Algebra ◽  
Free Algebras ◽  
Endomorphism Semigroup ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Group Of Automorphisms ◽  
Variety Of Associative Algebras

Let [Formula: see text] be the variety of associative algebras over a field K and A = K 〈x1,…, xn〉 be a free associative algebra in the variety [Formula: see text] freely generated by a set X = {x1,…, xn}, End A the semigroup of endomorphisms of A, and Aut End A the group of automorphisms of the semigroup End A. We prove that the group Aut End A is generated by semi-inner and mirror automorphisms of End A. A similar result is obtained for the automorphism group Aut [Formula: see text], where [Formula: see text] is the subcategory of finitely generated free algebras of the variety [Formula: see text]. The later result solves Problem 3.9 formulated in [17].

scholarly journals Presentations for Subrings and Subalgebras of Finite CO-Rank

2019 ◽  
Vol 71 (1) ◽  
pp. 53-71
Author(s):  
Peter Mayr ◽  
Nik Ruškuc
Keyword(s):  
Lie Algebra ◽  
Noetherian Ring ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Commutative Noetherian Ring ◽  
Rank 2 ◽  
Finitely Presented ◽  
Algebra Over A Field

Abstract Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented (resp. finitely generated) if and only if $B$ is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on $K$ and show that for finite generation it can be replaced by a weaker condition that the module $A/B$ be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension $1$ of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.

scholarly journals THE GROUP OF AUTOMORPHISMS OF THE SEMIGROUP OF ENDOMORPHISMS OF FREE COMMUTATIVE AND FREE ASSOCIATIVE ALGEBRAS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 941-949 ◽  
Author(s):  
A. BERZINS
Keyword(s):  
Associative Algebra ◽  
Free Associative Algebra ◽  
Associative Algebras ◽  
Group Of Automorphisms

Let W(X) be a free commutative or a free associative algebra. The group of automorphisms of the semigroup End (W(X)) is studied.

On the augmentation topology of automorphism groups of affine spaces and algebras

2018 ◽  
Vol 28 (08) ◽  
pp. 1449-1485 ◽  
Author(s):  
Alexei Kanel-Belov ◽  
Jie-Tai Yu ◽  
Andrey Elishev
Keyword(s):  
Automorphism Group ◽  
Associative Algebra ◽  
Automorphism Groups ◽  
Jacobian Conjecture ◽  
Free Associative Algebra ◽  
Topological Properties ◽  
Negative Solution ◽  
Base Field ◽  
Associative Algebras ◽  
Affine Spaces

We study topological properties of Ind-groups [Formula: see text] and [Formula: see text] of automorphisms of polynomial and free associative algebras via Ind-schemes, toric varieties, approximations, and singularities. We obtain a number of properties of [Formula: see text], where [Formula: see text] is the polynomial or free associative algebra over the base field [Formula: see text]. We prove that all Ind-scheme automorphisms of [Formula: see text] are inner for [Formula: see text], and all Ind-scheme automorphisms of [Formula: see text] are semi-inner. As an application, we prove that [Formula: see text] cannot be embedded into [Formula: see text] by the natural abelianization. In other words, the Automorphism Group Lifting Problem has a negative solution. We explore close connection between the above results and the Jacobian conjecture, as well as the Kanel-Belov–Kontsevich conjecture, and formulate the Jacobian conjecture for fields of any characteristic. We make use of results of Bodnarchuk and Rips, and we also consider automorphisms of tame groups preserving the origin and obtain a modification of said results in the tame setting.

scholarly journals Odd automorphisms of two generated braided free associative algebras

2020 ◽  
Vol 131 (2) ◽  
pp. 42-50
Author(s):  
R. Mutalip ◽  
◽  
A.S. Naurazbekova ◽  
Keyword(s):  
Associative Algebra ◽  
Free Associative Algebra ◽  
Arbitrary Field ◽  
Associative Algebras ◽  
Alpha Beta

It is proved that an endomorphism $\varphi$ of an braided free associative algebra in two generators over an arbitrary field $k$ with an involutive diagonal braiding $\tau = (- 1, -1, -1, -1)$ given by the rule $\varphi (x_1) = x_1, \, \varphi (x_2) = \alpha x_2 + \beta x^m_1,$ where $\alpha, \, \beta \in k, \, m $ is an odd number, is an odd automorphism. It is also proved that the linear endomorphism $\psi$ of this algebra is an automorphism if and only if $\psi$ is affine. It is shown that the group of all automorphisms of braided free associative algebra in two variables over an arbitrary field $ k $ with an involutive diagonal braiding $ \tau = (- 1, -1, -1, -1) $ coincides with the group of odd automorphisms of this algebra.

scholarly journals Grothendieck groups of twisted free associative algebras

1977 ◽  
Vol 18 (2) ◽  
pp. 193-196
Author(s):  
Koo-Guan Choo
Keyword(s):  
Associative Algebra ◽  
Associative Ring ◽  
Free Associative Algebra ◽  
Associative Algebras ◽  
Grothendieck Groups

Let R be an associative ring with identity, X a set of noncommuting variables, = {αx} x ∈ X a set of automorphisms αx of R and R {X} the -twisted free associative algebra on X over R. Let Y be another set of noncommuting variables, ℬ = {βy}y∈Y a set of automorphisms βy of R {X} and S = (R{X})ℬ {Y} the ℬ-twisted free associative algebra on Y over R{X}. Next, let X1 be a set of noncommuting variables, for each l = 1,2,…. We form the free associative algebra S1 = S{X1}on Xl over S and inductively, we form the free associative algebra Sl+1 = Sl{Xl+1} on Xl+1 over Sl, l = 1,2,….

scholarly journals Finite generation of Lie derived powers of associative algebras

2019 ◽  
Vol 18 (03) ◽  
pp. 1950059
Author(s):  
Adel Alahmadi ◽  
Hamed Alsulami
Keyword(s):  
Lie Algebras ◽  
Associative Algebra ◽  
Finite Collection ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Finite Generation ◽  
Nilpotent Elements ◽  
Algebra Over A Field

Let [Formula: see text] be an associative algebra over a field of characteristic [Formula: see text] that is generated by a finite collection of nilpotent elements. We prove that all Lie derived powers of [Formula: see text] are finitely generated Lie algebras.

scholarly journals TAME AUTOMORPHISMS FIXING A VARIABLE OF FREE ASSOCIATIVE ALGEBRAS OF RANK THREE

2007 ◽  
Vol 17 (05n06) ◽  
pp. 999-1011 ◽  
Author(s):  
VESSELIN DRENSKY ◽  
JIE-TAI YU
Keyword(s):  
Associative Algebra ◽  
Free Associative Algebra ◽  
Associative Algebras

We study automorphisms of the free associative algebra K〈x,y,z〉 over a field K which fix the variable z. We describe the structure of the group of z-tame automorphisms and derive algorithms which recognize z-tame automorphisms and z-tame coordinates.

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