Quasi-Direct Decompositions of Torsion-Free Abelian Groups of Infinite Rank.

Fuchs, L.; Viljoen, G.

doi:10.7146/math.scand.a-11484

Cited by 78 publications

(128 citation statements)

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“…The countability restriction was finally removed by Kaplansky [9]. Kulikov's argument was improved by Fuchs in [5], and then given two very smooth renditions by B. Charles, [3], and G. Kolettis,[12]. We have based our categorical version primarily on Kolettis's version of the argument, but we also borrow some of Charles's point of view.…”

Section: Corollarymentioning

confidence: 99%

Classification theory of abelian groups. I. Balanced projectives

Warfield¹

1976

Trans. Amer. Math. Soc.

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ABSTRACT. We introduce in this paper a class of Abelian groups which includes the torsion totally projective groups and those torsion-free groups which are direct sums of groups of rank one. Characterizations of the groups in this class are given, and a complete classification theorem, in terms of additive numerical invariants, is proved.By a height, we mean a formal product h = Tl p"^p\ where v(p) is either an ordinal or the symbol °°, and the product is taken over all primes p. If G is an Abelian group, we define hG = C\ pv(-p^G. A short exact sequence 0 -■*■ A -*■ B -> C -> 0 is balanced if for every height h, the sequence 0 -* hA -► hB -* hC -> 0 is exact. A group P is a balanced projective if it is projective with respect to all such sequences-i.e., for any such sequence, the induced map Hom(P, B) -► Hom(/>, C) is surjective. A torsion-free group is balanced projective if and and only if it is a direct sum of groups of rank one. A p-group is balanced projective if and only if it is the direct sum of a divisible group and a totally projective group (defined below or in [17], [23], [6]). We will prove a classification theorem for the balanced projectives which will include, as special cases, Baer's 1937 classification of the torsion-free balanced projectives [2], and Hill's classification of totally projective p-groups [7], [23]. We also prove that any balanced projective is a direct sum of groups of torsion-free rank at most one. (Most of these rank one groups are not direct sums of a torsion group and a torsion-free group, so the invariants we need cannot be obtained directly from Baer's invariants and the Ulm invariants.)An important tool in developing the theory is to work first in the local case-with modules over a discrete valuation ring. One then proves many of the results for an Abelian group M by referring to the localization M = M ® Zp

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Section: Corollarymentioning

confidence: 99%

Classification theory of abelian groups. I. Balanced projectives

Warfield¹

1976

Trans. Amer. Math. Soc.

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“…Hence png E K and q \pn. This contradiction implies that (G/K)q = 0 and hence Kis a-isotype in G (see [3,Lemma …”

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confidence: 99%

“…Introduction. All groups in this paper are abelian, we shall follow the notation and terminology of [3]. Let G be a group.…”

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confidence: 99%

Intersections of Γ-isotype subgroups in abelian groups

Bečvář¹

1982

Proc. Amer. Math. Soc.

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A subgroup H H of an abelian group G G is an intersection of isotype subgroups of G G if and only if, for each prime p p , if x + H x + H is a coset of order p p then there is another coset of order p p containing an element y y of order p p such that h p ∗ ( x ) ⩽ h p ∗ ( y ) h_p^* (x) \leqslant h_p^*(y) . A subgroup H H of G G is isotype in G G if and only if, for each prime p p , every coset x + H x + H of order p p contains an element y of order p p such that h p ∗ ( x ) h p ∗ ( y ) h_p^*(x)h_p^*(y)

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“…Most examples and results dealt with almost completely decomposable groups that are, by definition, finite extensions of finite rank completely decomposable groups, and are relatively accessible. Main references are [Jon57], [Jon59], [Cor61], [FL70], [BM80], [Bla83], [Yak89], [BY90], [YK93], [BM94], but, in some form or fashion, the topic is touched upon in almost every publication on almost completely decomposable groups. On the other hand, there are also uniqueness results, the most remarkable and general of these being the theorem of Faticoni and Schultz ( [FS96]; for a different proof see [MV97] or [Mad00, Section 10.4]), which says that an indecomposable decomposition of a group that is the extension of a completely decomposable group by a finite primary group is unique up to near-isomorphism.…”

Section: Introductionmentioning

confidence: 99%