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Let R be a commutative ring, M an R-module of finite length and let N be an Injective R-module with zero socle. Then why $ \text{Hom}_R(M, N) $ is zero?

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How does this question arise in your research? –  S. Carnahan Apr 21 '13 at 5:06
    
In a paper of Auslander(Isolated singularities and existence of AR sequences) What can be said if M and N are in R-mod. –  Max Apr 21 '13 at 5:37
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1 Answer

if some hom is non-zero, then the image is a module of finite length, which allows to find an irreducible submodule in $N$ giving a contradiction.

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