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This work presents a sample constructions of two algebras both with the idealof relations defined by a finite Gr\"obner basis. For the first algebra thequestion whether a given element is nilpotent is algorithmically unsolvable,for the second one the question whether a given element is a zero divisor isalgorithmically unsolvable. This gives a negative answer to questions raised byLatyshev.

Finite Gr\"obner basis algebra with unsolvable nilpotency problem and zero divisors problem