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ℵn-FREE MODULES OVER COMPLETE DISCRETE VALUATION DOMAINS WITH ALMOST TRIVIAL DUAL *
RÜDIGER GÖBEL, SAHARON SHELAH, LUTZ STRÜNGMANN
Journal:

Glasgow Mathematical Journal / Volume 55 / Issue 2 / May 2013

Published online by Cambridge University Press:

25 February 2013, pp. 369-380

Print publication:

May 2013
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A module M over a commutative ring R has an almost trivial dual if there is no homomorphism from M onto a free R-module of countable infinite rank. Using a new combinatorial principle (the ℵn-Black Box), which is provable in ordinary set theory, we show that for every natural number n, there exist arbitrarily large ℵn-free R-modules with almost trivial duals, when R is a complete discrete valuation domain. A corresponding result for torsion modules is also obtained.