Corner, A. L. S. (2008), "Groups of units of orders in Q-algebras", Models, modules and abelian groups, Walter de Gruyter, Berlin, pp. 9–61, doi:10.1515/9783110203035.9, MR2513226. See in particular the proof of Lemma H.4, p. 36, which uses this fact.
Blass, Andreas (1979), "Injectivity, projectivity, and the axiom of choice", Transactions of the American Mathematical Society, 255: 31–59, doi:10.1090/S0002-9947-1979-0542870-6, JSTOR1998165, MR0542870. For the connection to free objects, see Corollary 1.2. Example 7.1 provides a model of set theory without choice, and a non-free projective abelian group in this model that is a subgroup of a free abelian group , where is a set of atoms and is a finite integer. Blass writes that this model makes the use of choice essential in proving that every projective group is free; by the same reasoning it also shows that choice is essential in proving that subgroups of free groups are free.
Johnson, D. L. (1980), Topics in the Theory of Group Presentations, London Mathematical Society lecture note series, vol. 42, Cambridge University Press, p. 9, ISBN978-0-521-23108-4, MR0695161
Tolstykh, Vladimir (2005), "What does the automorphism group of a free abelian group A know about A?", in Blass, Andreas; Zhang, Yi (eds.), Logic and its Applications, Contemporary Mathematics, vol. 380, Providence, Rhode Island: American Mathematical Society, pp. 283–296, arXiv:math/0701752, doi:10.1090/conm/380/07117, MR2167584, S2CID18107280
Ayoub, Raymond G.; Ayoub, Christine (1969), "On the group ring of a finite abelian group", Bulletin of the Australian Mathematical Society, 1 (2): 245–261, doi:10.1017/S0004972700041496, MR0252526
Tolstykh, Vladimir (2005), "What does the automorphism group of a free abelian group A know about A?", in Blass, Andreas; Zhang, Yi (eds.), Logic and its Applications, Contemporary Mathematics, vol. 380, Providence, Rhode Island: American Mathematical Society, pp. 283–296, arXiv:math/0701752, doi:10.1090/conm/380/07117, MR2167584, S2CID18107280
Corner, A. L. S. (2008), "Groups of units of orders in Q-algebras", Models, modules and abelian groups, Walter de Gruyter, Berlin, pp. 9–61, doi:10.1515/9783110203035.9, MR2513226. See in particular the proof of Lemma H.4, p. 36, which uses this fact.
Corner, A. L. S. (2008), "Groups of units of orders in Q-algebras", Models, modules and abelian groups, Walter de Gruyter, Berlin, pp. 9–61, doi:10.1515/9783110203035.9, MR2513226. See in particular the proof of Lemma H.4, p. 36, which uses this fact.
Blass, Andreas (1979), "Injectivity, projectivity, and the axiom of choice", Transactions of the American Mathematical Society, 255: 31–59, doi:10.1090/S0002-9947-1979-0542870-6, JSTOR1998165, MR0542870. For the connection to free objects, see Corollary 1.2. Example 7.1 provides a model of set theory without choice, and a non-free projective abelian group in this model that is a subgroup of a free abelian group , where is a set of atoms and is a finite integer. Blass writes that this model makes the use of choice essential in proving that every projective group is free; by the same reasoning it also shows that choice is essential in proving that subgroups of free groups are free.
Tolstykh, Vladimir (2005), "What does the automorphism group of a free abelian group A know about A?", in Blass, Andreas; Zhang, Yi (eds.), Logic and its Applications, Contemporary Mathematics, vol. 380, Providence, Rhode Island: American Mathematical Society, pp. 283–296, arXiv:math/0701752, doi:10.1090/conm/380/07117, MR2167584, S2CID18107280
Ayoub, Raymond G.; Ayoub, Christine (1969), "On the group ring of a finite abelian group", Bulletin of the Australian Mathematical Society, 1 (2): 245–261, doi:10.1017/S0004972700041496, MR0252526
Norman, Christopher (2012), "1.3 Uniqueness of the Smith Normal Form", Finitely Generated Abelian Groups and Similarity of Matrices over a Field, Springer undergraduate mathematics series, Springer, pp. 32–43, Bibcode:2012fgag.book.....N, ISBN9781447127307
jstor.org
Blass, Andreas (1979), "Injectivity, projectivity, and the axiom of choice", Transactions of the American Mathematical Society, 255: 31–59, doi:10.1090/S0002-9947-1979-0542870-6, JSTOR1998165, MR0542870. For the connection to free objects, see Corollary 1.2. Example 7.1 provides a model of set theory without choice, and a non-free projective abelian group in this model that is a subgroup of a free abelian group , where is a set of atoms and is a finite integer. Blass writes that this model makes the use of choice essential in proving that every projective group is free; by the same reasoning it also shows that choice is essential in proving that subgroups of free groups are free.
Tolstykh, Vladimir (2005), "What does the automorphism group of a free abelian group A know about A?", in Blass, Andreas; Zhang, Yi (eds.), Logic and its Applications, Contemporary Mathematics, vol. 380, Providence, Rhode Island: American Mathematical Society, pp. 283–296, arXiv:math/0701752, doi:10.1090/conm/380/07117, MR2167584, S2CID18107280