I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
1996, VOLUME 2, NUMBER 2, PAGES 563-594
The model theory of divisible modules over a domain
V. A. Puninskaya
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A connected module over a commutative
a regular generic type iff it is divisible as a module over the
integral domain .
Given a divisible module over an integral
we identify a certain ring introduced by Facchini
as the ring of definable endomorphisms of .
strongly minimal, then either is a field and
vector space over , or 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 -modules for which
every primary component of the torsion submodule is artinian.
We also note that if a superstable module over a commutative
(with no additional structure) has a regular generic type, then the
of is an
If 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 -module and, by elementary
duality, a big (not finitely generated) pure-injective indecomposable
Last modified: March 19, 2005