Pythagorean triples in the unification theory of associative and commutative rings

Ruvim Lipyanski

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

All solutions of Pythagorean equation (P-equation) x12+ x22 = x32 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 languageEnglish
Pages (from-to)5137-5154
Number of pages18
JournalCommunications in Algebra
Volume30
Issue number11
DOIs
StatePublished - 1 Nov 2002

Keywords

  • Nilpotent rings
  • Pythagorean triples
  • Unification type

ASJC Scopus subject areas

  • Algebra and Number Theory

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