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1. • CommentRowNumber1.
   • CommentAuthorStephan A Spahn
   • CommentTimeMay 15th 2012
   • (edited May 15th 2012)
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   Author: Stephan A Spahn
   Format: MarkdownItexI added the characterization of a divisible abelian group as an injective object in the category of abelian groups to [[divisible group]].

   I added the characterization of a divisible abelian group as an injective object in the category of abelian groups to divisible group.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeMay 17th 2012
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   Author: zskoda
   Format: MarkdownItexThere is some basic result on this which Hyman Bass proved as an (undergrad I think) student, I vaguely recall, usually quoted in homological algebra textbooks. Is this very fact or something else ?

   There is some basic result on this which Hyman Bass proved as an (undergrad I think) student, I vaguely recall, usually quoted in homological algebra textbooks. Is this very fact or something else ?

3. • CommentRowNumber3.
   • CommentAuthorStephan A Spahn
   • CommentTimeMay 18th 2012
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   Author: Stephan A Spahn
   Format: MarkdownItexThe statement that an abelian group is divisible iff it is an injective object I found in Proposition 3.19 in Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York. I added this reference. Do you remember what the statement of Hyman Bass was?

   The statement that an abelian group is divisible iff it is an injective object I found in Proposition 3.19 in Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York. I added this reference.

   Do you remember what the statement of Hyman Bass was?

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeMay 18th 2012
   • (edited May 18th 2012)
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   Author: zskoda
   Format: MarkdownItexAfter I reminded myself through the references, I see the statement is rather different and not about the divisible groups but more general results on injective modules. __Every infinite direct sum of injective right $R$-modules is injective if and only if the ring $R$ is right Noetherian__. cf. Lam (1999) 3.46 wikipedia [injective module](http://en.wikipedia.org/wiki/Injective_module) and Theorem 0.1 in * Carl Faith, Elbert A. Walker, _Direct sum representations of injective modules_, J. Algebra __5__:2, 203--221 (1967) [pdf](http://sofia-new.nmsu.edu/~elbert/DirectSumRepsOfInjectives.pdf) The statement is often referred to as Bass' lemma when in that direction, and sometimes the iff statement as Bass-Papp theorem. One direction is due Cartan-Eilenberg (right Noetherianess implies the other side) while the converse is due Hyman Bass from his student days, and, independently, due * Z. Papp, _On algebraically closed modules_, Publ. Math. Debrecen __6__ (1959) 311--327 Unfortunately online access to this journal I can find only from vol. 39 ([here](http://www.math.klte.hu/publi/contents.php)) till current vol. 80.

   After I reminded myself through the references, I see the statement is rather different and not about the divisible groups but more general results on injective modules.

   Every infinite direct sum of injective right $R$-modules is injective if and only if the ring $R$ is right Noetherian.

   cf. Lam (1999) 3.46 wikipedia injective module and Theorem 0.1 in

   • Carl Faith, Elbert A. Walker, Direct sum representations of injective modules, J. Algebra 5:2, 203–221 (1967) pdf

   The statement is often referred to as Bass’ lemma when in that direction, and sometimes the iff statement as Bass-Papp theorem. One direction is due Cartan-Eilenberg (right Noetherianess implies the other side) while the converse is due Hyman Bass from his student days, and, independently, due

   • Z. Papp, On algebraically closed modules, Publ. Math. Debrecen 6 (1959) 311–327

   Unfortunately online access to this journal I can find only from vol. 39 (here) till current vol. 80.