Algorithmic problems in associative algebras
type: Item
instance of: scholarly article
title: Algorithmic problems in associative algebras
zbMATH Open document ID: 0851.16023
DOI: 10.1007/BF02261749
published in: Algebra and Logic
publication date: 1995-04-20T00:00:00Z
full work available at URL: https://eudml.org/doc/187658
review text: The author proves that the occurrence problem for free algebras in the variety of algebras obeying the identity \([x_1,x_2][x_3,x_4][x_5,x_6][x_7,x_8]=0\) is undecidable. The next result is that there is no algorithm determining whether or not a finite system of elements in a free associative algebra is algebraically dependent. And the last result is that there exists a f.g. associative algebra with undecidable word problem, such that its relation module is free. These undecidability results are proven by interpretation of two-tape Minsky machines. In the proof of the second result the cycles in these machines are studied and interpreted as dependences in algebras.
Mathematics Subject Classification ID: 16S15
Mathematics Subject Classification ID: 16R10
zbMATH DE Number: 738958
zbMATH Keywords: algorithmic problems
zbMATH Keywords: algebraic dependence
zbMATH Keywords: occurrence problem
zbMATH Keywords: free algebras
zbMATH Keywords: variety of algebras
zbMATH Keywords: undecidable word problem
zbMATH Keywords: relation modules
zbMATH Keywords: undecidability results
zbMATH Keywords: two-tape Minsky machines
author: Ualbai Umirbaev
MaRDI profile type: Publication
cites work: Q5821489
cites work: Subalgebras of Free Associative Algebras
cites work: Q4297725
cites work: Q4189352
cites work: Using Gröbner bases to determine algebra membership, split surjective algebra homomorphisms determine birational equivalence
cites work: Q5519136
cites work: The equality problem and free products of Lie algebras and of associative algebras
cites work: Q3831180
cites work: On the cohomology groups of an associative algebra
cites work: Q3909162
OpenAlex ID: W2026893279
description: scientific article; zbMATH DE number 738958