## Abstract

Suppose that F_{2,m} is a free 2-generated associative ring with the identity x^{m} = 0. In 1993 Zelmanov put the following question: is it true that the nilpotency degree of F_{2,m} has exponential growth? We give the defnitive answer to Zelmanov's question by showing that the nilpotency class of an l-generated associative algebra with the identity x^{d} = 0 is smaller than Ψ(d,d,l), where (equation) This result is a consequence of the following fact based on combinatorics of words. Let l, n and d ≥ n be positive integers. Then all words over an alphabet of cardinality l whose length is not less than Ψ(n,d,l) are either n-divisible or contain x^{d}; a word W is n-divisible if it can be represented in the form W = W _{0}W_{1} · · · W_{n} so that W_{1},., W_{n} are placed in lexicographically decreasing order. Our proof uses Dilworth's theorem (according to V. N. Latyshev's idea). We show that the set of not n-divisible words over an alphabet of cardinality l has height h < Φ(n, l) over the set of words of degree ≤ n-1, where 87 12 logc, n+48 Φ(n, l) = 2^{87}l · n^{12log} _{3}n+48 Bibliography: 40 titles.

Original language | English |
---|---|

Pages (from-to) | 534-553 |

Number of pages | 20 |

Journal | Sbornik Mathematics |

Volume | 203 |

Issue number | 4 |

DOIs | |

State | Published - 2012 |

Externally published | Yes |

## Keywords

- Burnside-type problems
- Dilworth theorem
- Shirshov theorem on height
- Word combinatorics
- n-divisibility