Example direct sum of injective modules is not injective

Asked Oct 03 '13 at 10:31

Active Apr 20 '15 at 12:48

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We knew that a direct product of injective modules is injective but a infinite direct sum need not be. I searched many article about this problem, and all most given a counter-example is: the Bass-Papp Theorem. So, is there a easy example about that, meaning, over $Z, Q...$

Thanks for regarding my question!

edited Aug 27 '14 at 19:29

user26857

52,094

asked Oct 03 '13 at 10:31

Rachel

1

http://mathoverflow.net/questions/103402/the-direct-sum-of-injective-modules-need-not-be-injective – Prahlad Vaidyanathan Oct 03 '13 at 10:36
A ring is (left) noetherian if and only if every direct sum of injective (left) modules is injective. – egreg Oct 03 '13 at 10:36
In particular over nice rings (Z,Q,...) there won't be a counterexample. – Martin Brandenburg Oct 03 '13 at 11:19
@Martin Brandenburg: thanks, that's all I need – Rachel Oct 03 '13 at 11:21
@Prahlad Vaidyanathan, egreg: thanks, I knew that before – Rachel Oct 03 '13 at 11:22

Example direct sum of injective modules is not injective

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