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Results in Mathematics

, 74:14 | Cite as

Forms of Choice in Ring Theory

  • Lorenz Halbeisen
  • Norbert Hungerbühler
  • Nir Lazarovich
  • Waltraud Lederle
  • Marc Lischka
  • Salome Schumacher
Open Access
Article
  • 16 Downloads

Abstract

We investigate the relationship between various choice principles and \(n\hbox {th}\)-root functions in rings. For example, we show that the Axiom of Choice is equivalent to the statement that every ring has a square-root function. Furthermore, we introduce a choice principle which implies that every integral domain has an \(n\hbox {th}\)-root function (for odd integers n), and introduce another choice principle which is equivalent to the Prime Ideal Theorem restricted to certain ideals. Finally, we investigate the dependencies between the two new choice principles and a choice principle for families of n-element sets.

Keywords

Square root functions in rings root functions in integral domains axiom of choice finite choice cycle choice bounded multiple choice consistency results 

Mathematics Subject Classification

03E25 13A99 

Notes

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Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsETH ZentrumZurichSwitzerland
  2. 2.Department of MathematicsTechnion – Israel Institute of TechnologyHaifaIsrael

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