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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Injectivity, projectivity, and the axiom of choice
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by Andreas Blass PDF
Trans. Amer. Math. Soc. 255 (1979), 31-59 Request permission

Abstract:

We study the connection between the axiom of choice and the principles of existence of enough projective and injective abelian groups. We also introduce a weak choice principle that says, roughly, that the axiom of choice is violated in only a set of different ways. This principle holds in all ordinary Fraenkel-Mostowski-Specker and Cohen models where choice fails, and it implies, among other things, that there are enough injective abelian groups. However, we construct an inner model of an Easton extension with no nontrivial injective abelian groups. In the presence of our weak choice principle, the existence of enough projective sets is as strong as the full axiom of choice, and the existence of enough free projective abelian groups is nearly as strong. We also prove that the axiom of choice is equivalent to “all free abelian groups are projective” and to “all divisible abelian groups are injective."
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Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 255 (1979), 31-59
  • MSC: Primary 04A25; Secondary 03E35, 20K99
  • DOI: https://doi.org/10.1090/S0002-9947-1979-0542870-6
  • MathSciNet review: 542870