Gauß' theorem for two submodules

Files, Steve T.; Göbel, Rüdiger

doi:10.1007/pl00004630

Cited by 11 publications

(8 citation statements)

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“…The following consequence of Corollary 4.6 has been noted in a number of recent papers [5,9,28,22,46]. It provides, in effect, a complete classification of finite-rank Butler groups with at most two critical types.…”

Section: Applications To Butler Groupsmentioning

confidence: 73%

“…Butler groups with a critical typeset of the form T n (an antichain of length n with a bottom element) for n = 2 are classified in [43] as direct sums of indecomposable T 2 -Butler groups of rank ≤ 2. This also follows from Rep 2 R-considerations in [5,28]. We will bring our machinery in Rep n R for n = 2, 3 and 4 to bear on T n -groups in the final section, obtaining group-theoretical results about Butler groups of various torsion-free ranks, both finite and infinite.…”

Section: Introductionmentioning

confidence: 93%

“…In recent accounts of representation theory and Butler groups (see [5,28]), R 2 -modules (F, F 0 , F 1 ) with pure submodules F 0 and F 1 have played a major role. It is worth pointing out how an important theorem concerning such representations Proof.…”

Section: Representations Over Pid's With Three Distinguished Submodulmentioning

confidence: 99%

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Representations over PID’s with three distinguished submodules

Files

Göbel

2000

Trans. Amer. Math. Soc.

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Abstract. Let R be a principal ideal domain. The R-representations with one distinguished submodule are classified by a theorem of Gauß in the case of finite rank, and by the "Stacked Bases Theorem" of Cohen and Gluck in the case of infinite rank. Results of Hill and Megibben carry this classification even further. The R-representations with two distinguished pure submodules have recently been classified by Arnold and Dugas in the finite-rank case, and by the authors for countable rank. Although wild representation type prevails for R-representations with three distinguished pure submodules, an extensive category of such objects was recently classified by Arnold and Dugas. We carry their groundbreaking work further, simplifying the proofs of their main results and applying new machinery to study the structure of finite-and infinite-rank representations with two, three, and four distinguished submodules. We also apply these results to the classification of Butler groups, a class of torsion-free abelian groups that has been the focus of many investigations over the last sixteen years.

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Section: Applications To Butler Groupsmentioning

confidence: 73%

Section: Introductionmentioning

confidence: 93%

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Representations over PID’s with three distinguished submodules

Files

Göbel

2000

Trans. Amer. Math. Soc.

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“…In 1969 Kaplansky's conjecture was proved to be true by J.M. The most important theorems were obtained by D. Arnold, M. Dugas (see [6]) and S. Files, R. GoÈ bel (see [7] and [8]). In the special case when B is ®nitely generated and R Z the result of Cohen and Gluck extends the classical and wellknown invariant factor theorem for ®nitely generated abelian groups (see [3]).…”

Section: Introductionmentioning

confidence: 99%

Stacked Bases for Countable Homogeneous Completely Decomposable Groups

Ould-Beddi

Strüngmann

2001

Communications in Algebra

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Let A X be two homogeneous completely decomposable torsion free abelian groups of the same type t and of countable rank such that the quotient X =A is a direct sum of torsion cyclic groups and a homogeneous completely decomposable summand of type t. Furthermore assume that A and X have a common direct summand of countable rank. We show that there exist stacked bases for A and X , i.e. there existThis proves a stacked bases theorem for pairs of homogeneous completely decomposable torsion free abelian groups of countable rank and the same type with a large common summand.

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“…In some sense, this paper was a prelude to Rüdiger's importation into Abelian group theory of many of the ideas arising in set theory and infinite combinatorics, which, as noted above, became one of his characteristics. The Baer-Specker group and the wonderful complexity of its set of subgroups were a topic of constant interest to Rüdiger, and he had many subsequent works in this area-see, for example, [24,25,31,84,85,100,103,107,119].…”

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confidence: 99%

Groups, Modules, and Model Theory - Surveys and Recent Developments

Droste¹,

Fuchs²,

Goldsmith³

et al. 2017

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Gauß' theorem for two submodules

Cited by 11 publications

References 55 publications

Representations over PID’s with three distinguished submodules

Representations over PID’s with three distinguished submodules

Stacked Bases for Countable Homogeneous Completely Decomposable Groups

Groups, Modules, and Model Theory - Surveys and Recent Developments

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