A solution to the splitting mixed group problem of Baer

Griffith, Phillip

doi:10.1090/s0002-9947-1969-0238957-1

Cited by 54 publications

(9 citation statements)

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“…The proof of the existence of subgroups A a9 a < μ, satisfying conditions (0) - (7) is, of course, by transfinite induction, and it employs the back-and-forth technique utilized in [1], [2], [3], [4], [5], [6] and a number of other recent papers. The reader who is well acquainted with these papers may need no further details concerning the existence of the subgroups A a satisfying conditions (0) - (7); however, we shall include a brief outline of the proof.…”

Section: G Is An ^ω-Group Then G Is Necessarily a Direct Sum Of Cyclmentioning

confidence: 99%

Primary groups whose subgroups of smaller cardinality are direct sums of cyclic groups

Hill¹

1972

Pacific J. Math.

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Section: G Is An ^ω-Group Then G Is Necessarily a Direct Sum Of Cyclmentioning

confidence: 99%

Primary groups whose subgroups of smaller cardinality are direct sums of cyclic groups

Hill¹

1972

Pacific J. Math.

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“…This shows that under V = L, countable torsion groups are enough to test whether a torsion-free group is Butler. This compares with P. Eklof's analysis of the splitting problem of mixed groups [12]. The Baer splitting problem, solved by Griffith [14], becomes undecidable in standard ZFC set theory if one restricts the class of torsion groups to the countable ones. The same happens here for the splitting of balanced extensions.…”

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confidence: 97%

Infinite Rank Butler Groups

Dugas¹,

Rangaswamy²

1988

Transactions of the American Mathematical Society

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Abstract. A torsion-free abelian group G is said to be a Butler group if Bext(C, T) = 0 for all torsion groups T. It is shown that Butler groups of finite rank satisfy what we call the torsion extension property (T.E.P.). A crucial result is that a countable Butler group G satisfies the T.E.P. over a pure subgroup H if and only if H is decent in G in the sense of Albrecht and Hill. A subclass of the Butler groups are the so-called B2-groups. An important question left open by Arnold, Bican, Salce, and others is whether every Butler group is a ^-group. We show under ( V = L) that this is indeed the case for Butler groups of rank Nt. On the other hand it is shown that, under ZFC, it is undecidable whether a group B for which Bext( B, T) = 0 for all countable torsion groups T is indeed a B2-group.

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“…It took quite a long time to prove that these notions actually coincide with the well-known notions of a free group and a projective module, respectively. Countable Baer groups were shown to be free already in 1936 by Baer [8], but the arbitrary ones only in 1969 by Griffith [16]. The projectivity of the classical Baer modules was also shown in two steps spreading over decades, however, in a different order.…”

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Baer and Mittag-Leffler modules over tame hereditary algebras

2009

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Abstract. We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if Ext 1 R (M, T ) = 0 for all torsion modules T , and M is Mittag-Leffler in case the canonical mapWe show that a module M is Baer iff M is p-filtered where p is the preprojective component of the tame hereditary algebra R. We apply this to prove that the universal localization of a Baer module is projective in case we localize with respect to a complete tube. Using infinite dimensional tilting theory we then obtain a structure result showing that Baer modules are more complex then the (infinite dimensional) preprojective modules. In the final section, we give a complete classification of the Mittag-Leffler modules.Since the fundamental work of Ringel [29], the study of infinite dimensional modules has become one of the challenging tasks of the representation theory of finite dimensional hereditary algebras. In the present paper, we consider in detail two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones.Besides proving structure results, we investigate further the surprising analogy with modules over Dedekind domains discovered in [29]. Indeed, the current progress relies heavily on applications of the recent powerful set-theoretic and homological methods developed originally for modules over domains, cf.[3] and [15].Baer modules can be defined in a rather general setting [26]: Let R be a ring and T be a torsion class in Mod−R. A module M ∈ Mod−R is a Baer module for T provided that Ext 1 R (M, T ) = 0 for all T ∈ T . In the particular case when R = Z, and more generally when R is a integral domain and T is the class of all torsion R-modules, we obtain thus the classical notions of a Baer group [8] and a Baer module [17]. It took quite a long time to prove that these notions actually coincide with the well-known notions of a free group and a projective module, respectively. Countable Baer groups were shown to be free already in 1936 by Baer [8], but the arbitrary ones only in 1969 by Griffith [16]. The projectivity of the classical Baer modules was also shown in two steps spreading over decades, however, in a different order. First, a reduction to countably presented modules was proved by set-theoretic methods by Eklof, Fuchs

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A solution to the splitting mixed group problem of Baer

Cited by 54 publications

References 9 publications

Primary groups whose subgroups of smaller cardinality are direct sums of cyclic groups

Primary groups whose subgroups of smaller cardinality are direct sums of cyclic groups

Infinite Rank Butler Groups

Baer and Mittag-Leffler modules over tame hereditary algebras

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