Characterizing a Class of Warfield Modules by Relation Arrays

Warfield modules and simply presented modules are considered whose torsion submodule is a direct sum of cyclics. A correspondence between Warfield modules and completely decomposable groups is established by showing that such a mixed module G is Warfield if and only if Gap w G is simply presented and p w G is completely decomposable. We give an example to show that this result cannot be extended to ordinals s b w. We also connect with the work of Nunke to characterize modules G whose zeroth Ulm factors are torsion in terms of Gap s G simply presented and p s G completely decomposable.Introduction. In this paper we investigate Warfield modules and simply presented modules whose torsion submodules are a direct sum of cyclics. We show a correspondence between Warfield modules and completely decomposable modules by showing that such mixed modules G are Warfield if and only if p s G is completely decomposable and Gap s G is Warfield for s bigger or equal to w. This is a strengthening of a result of Hill and Megibben [3]. Considering only the ordinal w we obtain a stronger result, namely, a module G is Warfield if and only if its first Ulm submodule p w G is completely decomposable and its zeroth Ulm factor Gap w G is simply presented. We show that for modules G with torsion zeroth Ulm factor the last result can be extended to ordinals s b w. However this breaks down if the module contains indicators of both the finite and w-type. Indeed we explicitly describe such a module G of torsion-free rank two which is Warfield but where Gap w1 G is not simply presented.

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“…[6,7,8,9]. As in [6] we denote by h 1 the subclass of h consisting of the modules of torsion-free rank one.…”

Section: Preliminariesmentioning

confidence: 99%