The model theory of divisible modules over a domain

I. Herzog ; V. A. Puninskaya

Fundamentalʹnaâ i prikladnaâ matematika, Tome 2 (1996) no. 2, pp. 563-594.

Résumé

A connected module $M$ over a commutative ring $R$ has a regular generic type iff it is divisible as a module over the integral domain $R/\!\operatorname{ann}_R (M)$. Given a divisible module $M$ over an integral domain $R$, we identify a certain ring $R(M)$ introduced by Facchini as the ring of definable endomorphisms of $M$. If $M$ is strongly minimal, then either $R(M)$ is a field and $M$ an infinite vector space over $R(M)$, or $R(M)$ is a 1-dimensional noetherian domain all of whose simple modules are finite. Matlis' theory of divisible modules over such a ring is applied to characterize the remaining strongly minimal modules as precisely those divisible $R(M)$-modules for which every primary component of the torsion submodule is artinian. We also note that if a superstable module $M$ over a commutative ring $R$ (with no additional structure) has a regular generic type, then the $U$-rank of $M$ is an indecomposable ordinal. If $R$ is a complete local 1-dimensional noetherian domain that is not of Cohen-Macaulay finite representation type, we apply Auslander's theory of almost-split sequences and the compactness of the Ziegler Spectrum to produce a big (non-artinian) torsion divisible pure-injective indecomposable $R$-module and, by elementary duality, a big (not finitely generated) pure-injective indecomposable Cohen-Macaulay $R$-module.