Absolutely Indecomposable Modules

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 a precise meaning, namely being smaller than the first ω-Erdős cardinal defined below. We will first apply a result on large rigid valuated trees with a similar property established by Shelah [26] in 1982, and will prove the existence of related ‘Rω-modules’ (R-modules with countably many distinguished submodules) and finally pass to R-modules. The passage through Rω-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 closes a gap in [12, 11], provides a good starting point for [16] and gives a new construction of indecomposable modules in general using a counting argument.