1996, VOLUME 2, NUMBER 2, PAGES 563-594

The model theory of divisible modules over a domain

I. Herzog
V. A. Puninskaya


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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/annR(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.

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Last modified: March 19, 2005