Let $R$ be a commutative ring and let $M$ and $N$ be two $R$-modules. Suppose that for every $R$-module $P$, the modules $Hom\_R(M,P)$ and $Hom\_R(N,P)$ are isomorphic. Is it true that $M$ and $N$ are isomorphic?



Are there two non-isomorphic modules such that all the Hom-sets are isomorphic?
[K. Bongartz, "A generalization of a theorem of Auslander"](http://blms.oxfordjournals.org/content/21/3/255.full.pdf+html): 



> 
> Let R be a commutative ring and A an abelian R-linear category such
>  that each morphism set in A has finite length as an R-module. Let C be a full
>  subcategory of A closed under direct sums and kernels. Then two objects M and N of
>  C are isomorphic if and only if the lengths of Hom(M, X) and Hom(N, X) coincide for
>  all X in C.
> 
> 
> 


