Finite Gröbner basis algebras with unsolvable nilpotency problem and zero divisors problem

Ilya Ivanov-Pogodaev; Sergey Malev

doi:10.1016/j.jalgebra.2018.02.035

Finite Gröbner basis algebras with unsolvable nilpotency problem and zero divisors problem

Ilya Ivanov-Pogodaev
, Sergey Malev

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

This work presents a sample construction of two algebras both with the ideal of relations defined by a finite Gröbner basis. For the first algebra the question whether a given element is nilpotent is algorithmically unsolvable, for the second one the question whether a given element is a zero divisor is algorithmically unsolvable. This gives a negative answer to questions raised by Latyshev.

Original language	English
Pages (from-to)	575-588
Number of pages	14
Journal	Journal of Algebra
Volume	508
DOIs	https://doi.org/10.1016/j.jalgebra.2018.02.035
State	Published - 15 Aug 2018
Externally published	Yes

Keywords

Algorithmic unsolvability
Finitely presented algebras
Finitely presented rings
Finitely presented semigroups
Noncommutative Gröbner basis
Turing machine

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10.1016/j.jalgebra.2018.02.035

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abstract = "This work presents a sample construction of two algebras both with the ideal of relations defined by a finite Gr{\"o}bner basis. For the first algebra the question whether a given element is nilpotent is algorithmically unsolvable, for the second one the question whether a given element is a zero divisor is algorithmically unsolvable. This gives a negative answer to questions raised by Latyshev.",

keywords = "Algorithmic unsolvability, Finitely presented algebras, Finitely presented rings, Finitely presented semigroups, Noncommutative Gr{\"o}bner basis, Turing machine",

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AU - Malev, Sergey

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N2 - This work presents a sample construction of two algebras both with the ideal of relations defined by a finite Gröbner basis. For the first algebra the question whether a given element is nilpotent is algorithmically unsolvable, for the second one the question whether a given element is a zero divisor is algorithmically unsolvable. This gives a negative answer to questions raised by Latyshev.

AB - This work presents a sample construction of two algebras both with the ideal of relations defined by a finite Gröbner basis. For the first algebra the question whether a given element is nilpotent is algorithmically unsolvable, for the second one the question whether a given element is a zero divisor is algorithmically unsolvable. This gives a negative answer to questions raised by Latyshev.

KW - Algorithmic unsolvability

KW - Finitely presented algebras

KW - Finitely presented rings

KW - Finitely presented semigroups

KW - Noncommutative Gröbner basis

KW - Turing machine

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ER -

Finite Gröbner basis algebras with unsolvable nilpotency problem and zero divisors problem

Abstract

Keywords

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