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I am searching for counterexamples to show that $\operatorname{Hom}_R(\prod_i M_i,N)$ may not be isomorphic to $\prod_i \operatorname{Hom}_R(M_i,N)$ or $\bigoplus_i\operatorname{Hom}_R(M_i,N)$ as $\mathbb Z$- modules when $R$ is a ring, and $M_i$ and $N$ are $R$-modules.

I chose $\mathbb Q$ or $\mathbb Z_n$ for $M_i$ and $N$ alternatively, and used some facts about injectiveness of $\mathbb Q$ but could not arrive at any good results. Thanks in advance!

karparvar
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  • Take all of them to be $\mathbb{Q}$. Then all three homs are vector spaces over $\mathbb{Q}$ of different dimensions (assuming the axiom of choice). You need to use the result cited here twice: https://mathoverflow.net/questions/268136/how-to-construct-a-basis-for-the-dual-space-of-an-infinite-dimensional-vector-sp – Qiaochu Yuan Aug 07 '24 at 07:31
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    A counterexample for $\operatorname{Hom}_R(\prod_i M_i,N) \cong \prod_i \operatorname{Hom}_R(M_i,N)$ is here. – azif00 Aug 07 '24 at 07:32

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