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

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.

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Intermediate growth in finitely presented algebras

International Journal of Algebra and Computation ◽

10.1142/s0218196717500205 ◽

2017 ◽

Vol 27 (04) ◽

pp. 391-401 ◽

Cited By ~ 1

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.

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

International Journal of Algebra and Computation ◽

10.1142/s0218196711006649 ◽

2011 ◽

Vol 21 (08) ◽

pp. 1339-1355 ◽

Cited By ~ 4

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).

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On some infinitely presented associative algebras

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700015068 ◽

1973 ◽

Vol 16 (3) ◽

pp. 290-293 ◽

Cited By ~ 10

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.

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