Absolutely Indecomposable Modules

TL;DR

This paper proves the existence of large, absolutely indecomposable abelian groups and modules over certain rings, using advanced set theory and linear algebra techniques to construct such modules with stable endomorphism rings.

Contribution

It introduces a new method for constructing absolutely indecomposable modules over large classes of rings, extending previous results with transparent linear algebra arguments.

Findings

Existence of large absolutely indecomposable abelian groups.

Construction of R_omega-modules with stable endomorphism rings.

Application of Shelah's rigid trees to module theory.

Abstract

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.