### Absolutely Indecomposable Modules

by Goebel and Shelah. [GbSh:880]

Proc American Math Soc, 2007

A module is called absolutely indecomposable if it is
directly indecomposable in every generic extension of the
universe. We want to show the existence of large abelian groups that
are absolutely indecomposable. This will follow from a more general
result about R-modules over a large class of commutative rings R
with endomorphism ring R which remains the same when passing to a
generic extension of the universe. It turns out that `large' in this
context has the precise meaning, namely being smaller then the
first omega-Erdos cardinal defined below. We will first apply
result on large rigid trees with a similar property established by
Shelah in 1982, and will prove the existence of related
` R_omega-modules' (R-modules with countably many distinguished
submodules) and finally pass to R-modules. The passage through
R_omega-modules has the great advantage that the proofs become
very transparent essentially using a few `linear algebra' arguments
accessible also for graduate students. The result gives a new
construction of indecomposable modules in general using a counting
argument.

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