Finite Gr\"obner basis algebra with unsolvable nilpotency problem and   zero divisors problem
  This work presents a sample constructions of two algebras both with the ideal of relations defined by a finite Gr\"obner 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. 
Ilya Ivanov-Pogodaev, Sergey Malev
math.RA