Continuing, a bit. I see stuff about $Hom$ and $\otimes$ being related through adjoint functors -- so is the Hom interpretation of tensors more durable than the idea of "generalized dual vectors" i.e. multilinear mappings into the ground ring?
So I believe there is a (natural) isomorphism between $L \otimes M$ and $Hom(L,M)$ for $R$-modules? More specifically, for each $L$, that would be a natural isomorphism between the functors $L \otimes \_$ and $Hom(L, \_)$ (or is it reversed, $Hom(\_,L)?$, which would assign a specific isomorphism between $L \otimes M$ and $Hom(L,M)$ to each $L$.
Can we give a specific construction for such isomorphisms? In message 16, I gave what I believe are the defining parameters for this isomorphism between $Z_a \otimes Z_b$ and $Hom(Z_a,Z_b)$.
Can we give a construction for this isomorphism, for the case of general $Z$-modules, or finite $Z$-modules.
For the latter, we could use the nice theorem, that any finite (or even finitely genenerated) $Z$-module is a direct sum of cyclic groups.
In the world of vector spaces, this isomorphism is a no-brainer: for vectors $v \in L$, $w \in M$, we could associate the following member of $Hom(L,M)$ with $v \otimes w$: $v^T(\_) * w$, where $v^T$ is the dual to $v$, relative to some basis for $L$.
That last example indicates that there will be many such isomorphisms, as it depends on the choice of a basis.
So in the above text, I am not asking for _the_ isomorphism, but just some isomorphism.