Example of divisible module that is not injective

Question

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

see [here](https://books.google.com/books?id=r9VoYbk-8c4C&pg=PA71&lpg=PA71&dq=divisible+module+not+injective&source=bl&ots=Gm3m5yNMtY&sig=e42A0YVxtrOwmm3c8c-eiXV-7wM&hl=en&sa=X&ei=HU3eVNThIoG3UufmgcAL&ved=0CEwQ6AEwBg#v=onepage&q=divisible%20module%20not%20injective&f=false) — user 1, Feb 13 '15 at 19:24

score 1 · Accepted Answer · answered Feb 13 '15 at 20:11

1

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

answered Feb 13 '15 at 20:11

user 1

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](http://math.stackexchange.com/questions/379248/disprove-injectivity). – 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](http://math.stackexchange.com/questions/379248/divisible-module-which-is-not-injective) 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

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