Classification theory of abelian groups, II: Local theory

Warfield, R. B.

doi:10.1007/bfb0090545

Cited by 24 publications

(14 citation statements)

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“…A simply presented module is a direct sum of submodules of torsion-free rank at most one. This was shown in [4, Lemma 1] and [13,Lemma 2.2]. Those modules which are direct summands of simply presented modules are called Warfield modules.…”

Section: Preliminariesmentioning

confidence: 97%

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Characterizing a class of Warfield modules by Ulm submodules and Ulm factors

2001

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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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Section: Preliminariesmentioning

confidence: 97%

“…In the sequel we will use this latter fact several times. As in [13], we call a module simply presented if it can be defined in terms of generators and relations in such a way that the only relations are of the form px y or px 0. A simply presented module is a direct sum of submodules of torsion-free rank at most one.…”

Section: Preliminariesmentioning

confidence: 99%

Characterizing a class of Warfield modules by Ulm submodules and Ulm factors

2001

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“…Warfield groups were introduced by Warfield in [7], where he defined numerical invariants, later named Warfield invariants, for this class of groups. He used these invariants to prove a generalized version of Ulm's theorem; namely, two Warfield groups are isomorphic if and only if they have the same Ulm and Warfield invariants.…”

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confidence: 99%

“…Some of these are already known. For example, totally projective p-groups can be characterized as simply presented p-groups, while Warfield groups are summands of simply presented groups [7]. Also, totally projective p-groups are the projectives relative to balanced exact sequences, and Warfield groups are the projectives relative to the sequentially pure exact sequences [7].…”

mentioning

confidence: 99%

“…For example, totally projective p-groups can be characterized as simply presented p-groups, while Warfield groups are summands of simply presented groups [7]. Also, totally projective p-groups are the projectives relative to balanced exact sequences, and Warfield groups are the projectives relative to the sequentially pure exact sequences [7]. The purpose of this paper is to present another characterization of Warfield groups; one which is the analogue of the description of totally projective p-groups as groups with a nice composition series.…”

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