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Assume $R$ is a graded ring and $M$ and $N$ graded modules. Denote by $^*\mathrm{Hom}_R(M,N)$ the set of all homogeneous $R$-linear maps from $M$ to $N$. How can I prove that if $M$ is finitely generated then $^*\mathrm{Hom}_R(M,N)=\mathrm{Hom}_R(M,N)$? Do you have a counterexample of a not finitely generated module $M$ such that this equality doesn't hold?

Chris
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