Tensor products isomorphic to hom-sets with a structure

Question

In which cases the tensor product of objects, say A and B, is (isomorphic with) the objects with the carrier set Hom(A,B) and a corresponding structure?

Answer 1

Okay, here's my attempt.

If $R$ is a principal ideal domain, $M$ and $N$ finitely-generated $R$-modules, then we can decompose $M$ and $N$ into direct sums $M=\oplus_k R/{I_k}$ and $N = \oplus_l R/{J_l}$.

We have $R/I \otimes R/J = R/(I+J)$, but $\operatorname{Hom}(R/I, R/J) = (J:I)/(I+J)$. So there is a natural (for cyclic modules only!) inclusion $\operatorname{Hom}(R/I, R/J) \to R/I \otimes R/J$, which is surjective if and only if $(J:I) = R$, i.e. $I\subset J$.

In particular, $M$ and $N$ are isomorphic whenever $I_k \subset J_l$ for all $k$ and all $l$. In some sense, we have this property when $M$ is "more free" than $N$, e.g. $M=\mathbb{Z}/6$, $N=\mathbb{Z}/2$.

So at the very least, we have a natural-ish (depends on the decomposition) isomorphism $M \otimes N \to \operatorname{Hom}(M,N)$ whenever $M$ is free, and $N$ is finitely generated.

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The above applies to vector spaces, and to vector space endomorphisms, which are $k[T]$-modules. In the latter, something like the following is the case: endomorphisms $T,T'$ should have a natural-ish isomorphism $T \otimes T' \to \operatorname{Hom}(T,T')$ whenever the characteristic polynomial of $T'$ divides the minimal polynomial of $T$.

This should shed some light on the case of graphs, which can be thought of as endomorphisms, even for directed multigraphs (send a basis vector $v_e$ to the sum of $v_{e'}$ over outgoing arrows $e\to e'$). I believe that the tensor product is the same, in which case it should be enough to look at minimal and characteristic polynomials of the adjacency matrices. If we restrict to connected graphs then I think the modules are cyclic, and we can even make this natural.