Algorithmically insoluble problems about finitely presented solvable groups, lie, and associative algebras

1984 ◽  
pp. 1-14
Author(s):  
Gilbert Baumslag
Keyword(s):  
Solvable Groups ◽  
Associative Algebras ◽  
Finitely Presented

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.

Intermediate growth in finitely presented algebras

2017 ◽  
Vol 27 (04) ◽  
pp. 391-401 ◽  
Author(s):  
Dilber Koçak
Keyword(s):  
Lie Algebras ◽  
Associative Algebras ◽  
Intermediate Growth ◽  
Type Formula ◽  
Growth Types ◽  
Finitely Presented

For any integer [Formula: see text], we construct examples of finitely presented associative algebras over a field of characteristic [Formula: see text] with intermediate growth of type [Formula: see text]. We produce these examples by computing the growth types of some finitely presented metabelian Lie algebras.

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 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.

Representations of Solvable Groups

1993 ◽  
Author(s):  
Olaf Manz ◽  
Thomas R. Wolf
Keyword(s):  
Solvable Groups
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