TY - JOUR

T1 - Finite Gröbner basis algebras with unsolvable nilpotency problem and zero divisors problem

AU - Ivanov-Pogodaev, Ilya

AU - Malev, Sergey

N1 - Publisher Copyright:
© 2018 Elsevier Inc.

PY - 2018/8/15

Y1 - 2018/8/15

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

UR - http://www.scopus.com/inward/record.url?scp=85047934358&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2018.02.035

DO - 10.1016/j.jalgebra.2018.02.035

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AN - SCOPUS:85047934358

SN - 0021-8693

VL - 508

SP - 575

EP - 588

JO - Journal of Algebra

JF - Journal of Algebra

ER -