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On strongly primary monoids and domains.

Alfred Geroldinger1, Moshe Roitman2

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Summary

This study explores properties of primary and strongly primary domains, proving that domains with a vanishing conductor are finite, which implies strongly primary domains are locally tame.

Keywords:
13A0513F0520M13Local tamenessone-dimensional local domainsprimary monoidssets of distancessets of lengths

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Area of Science:

  • Commutative algebra
  • Ring theory
  • Abstract algebra

Background:

  • Primary and strongly primary domains are defined based on dimensionality, locality, and ideal properties.
  • One-dimensional local Mori domains are a subset of strongly primary domains.
  • The concept of 'tame' domains relates to the factorization of elements into irreducibles.

Purpose of the Study:

  • To investigate the relationship between the vanishing conductor and the finiteness of irreducible element factorizations in domains.
  • To establish that strongly primary domains are locally tame.
  • To determine the conditions for a domain to be globally tame and answer a specific open problem.

Main Methods:

  • The study utilizes abstract algebraic techniques, focusing on properties of integral domains and monoids.
  • Key concepts include the conductor ideal, irreducible elements, and factorization properties.
  • The research extends results from domains to more general structures like commutative monoids.

Main Results:

  • If the conductor of a domain R vanishes, then R is finite, meaning every nonzero nonunit is a product of at most k irreducible elements.
  • Every strongly primary domain is proven to be locally tame.
  • A domain R is globally tame if and only if its conductor vanishes.

Conclusions:

  • The vanishing conductor is a crucial condition for finiteness and tameness in domains.
  • Strongly primary domains possess the property of being locally tame.
  • The research confirms a conjecture regarding globally tame domains, specifically answering Problem 38 from Cahen et al.