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.
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Blass, A.: Existence of bases implies the axiom of choice. In: Baumgartner, J.E., Martin, D.A., Shelah, S. (eds.) Axiomatic Set Theory, Contemporary Mathematics, vol. 31, pp. 31–33. American Mathematical Society, Providence (1984)
Halbeisen, L.J.: Combinatorial Set Theory, with a Gentle Introduction to Forcing. Springer Monographs in Mathematics, 2nd edn. Springer, London (2017)
Halpern, J.D.: Bases in vector spaces and the axiom of choice. Proc. Am. Math. Soc. 17, 670–673 (1966)
Hodges, W.: Krull implies Zorn. J. Lond. Math. Soc 19(2), 285–287 (1979)
Howard, P., Rubin, J.E.: Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, vol. 59. American Mathematical Society, Providence, RI (1998)
Jech, T.J.: The Axiom of Choice. Studies in Logic and the Foundations of Mathematics, vol. 75. North-Holland Publishing Co., Amercan Elsevier Publishing Co., Inc., Amsterdam, New York (1973)
Krull, W.: Idealtheorie in Ringen ohne Endlichkeitsbedingung. Math. Ann. 101, 729–744 (1929)
Läuchli, H.: Coloring infinite graphs and the Boolean prime ideal theorem. Isr. J. Math. 9, 422–429 (1971)
Pincus, D.: Zermelo-Fraenkel consistency results by Fraenkel-Mostowski methods. J. Symb. Log. 37, 721–743 (1972)
Stanley, R.P.: Zero square rings. Pac. J. Math. 30, 811–824 (1969)
Zuckerman, M.M.: Choosing \(l\)-element subsets of \(n\)-element sets. Pac. J. Math. 96, 247–250 (1981)
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Salome Schumacher: Partially supported by SNF grant 200021_178851.
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Halbeisen, L., Hungerbühler, N., Lazarovich, N. et al. Forms of Choice in Ring Theory. Results Math 74, 14 (2019). https://doi.org/10.1007/s00025-018-0935-1
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DOI: https://doi.org/10.1007/s00025-018-0935-1
Keywords
- Square root functions in rings
- root functions in integral domains
- axiom of choice
- finite choice
- cycle choice
- bounded multiple choice
- consistency results