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I know and can prove that $\operatorname{Ext}_Z^1(\mathbb{Z}/p\mathbb{Z},A) \simeq A/pA$. Does similar formula work for more general rings, such as Dedekind domains and their ideals, i.e. $\operatorname{Ext}_R^1(R/I,M) \simeq M/IM$ ? It seems that the original proof needs the principality of ideals, but I am unable to prove it otherwise or to find some counterexample.

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I got some very nice answers and examples regarding this question on mathoverflow: link. – Fred.Fred Aug 9 '12 at 8:38

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