## Abstract

All solutions of Pythagorean equation (P-equation) x_{1}^{2}+ x_{2}^{2} = x_{3}^{2} in relatively free rings of varieties of n-nilpotent associative or commutative-associative rings (n = 3,4) are described. In particular, it is shown that Pythagorean equation has no minimal and complete set of solutions in free rings of such varieties. This implies that unification type of these varieties is nullary. It is shown that unification type of P-equation in varieties of associative rings and commutative-associative rings without unit is not finitary. Hence, the unification type of these varieties is also not finitary. We show also that the variety of commutative-associative 3 (or 4)-nilpotent rings of characteristic 2 has nullary unification type.

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

Pages (from-to) | 5137-5154 |

Number of pages | 18 |

Journal | Communications in Algebra |

Volume | 30 |

Issue number | 11 |

DOIs | |

State | Published - 1 Nov 2002 |

## Keywords

- Nilpotent rings
- Pythagorean triples
- Unification type

## ASJC Scopus subject areas

- Algebra and Number Theory