  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.
