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 -