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We know that every injective module is divisible, but I can't find an example of divisible module such that it is not injective.

user26857
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pink floyd
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1 Answers1

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Hint
Joseph Rotman in the book An Introduction to Homological Algebra says

"a domain $R$ is a dedekind ring if and only if every divisible module is injective" (Theorem 4.24)

so you can consider a domain that is not a dedekind ring

user 1
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  • This answer doesn't provide a concrete example. (In fact, it doesn't provide anything, because a domain which is not Dedekind can have divisible modules which are injective.) A concrete example can be found here. – user26857 Feb 13 '15 at 22:03
  • @user26857. right; i doubt if op wanted concrete example(this answer is only a hint in this case) or wanted to sure that there exists some examples and sure converse of "every injective module is divisible" is not right. thanks you op for acceptation. BTW thank you for this link – user 1 Feb 14 '15 at 06:57
  • "but I can't find an example of..." – user26857 Feb 14 '15 at 07:55
  • as you see this is a Hint. it suggest that one can look for quotient of injective modules (in a domain that is not a dedekind ring), since the proof says there exist at list one quotient of an injective module such that is divisible but not injective. BTW, i agree that this answer does not give an explicit example. – user 1 Feb 14 '15 at 16:27