Warfield modules

Hunter, Roger; Richman, Fred; Walker, Elbert A.

doi:10.1007/bfb0068191

Cited by 28 publications

(17 citation statements)

References 8 publications

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“…Let x = a + b + c. Although it is not altogether obvious, (x) is quasi-sequentially nice in G but not knice. Lest the reader then think that Theorem 11 in [8] has a hypothesis genuinely weaker than our Theorem 3.4, we hasten to point out that it is easy to show that if N is quasi-sequentially nice in G with G/N a fc-module, then N is in fact knice in G.…”

Section: Warfield Modulesmentioning

confidence: 99%

“…As quasi-sequentially nice submodules have entered into the study of Warfield modules not only in [12] but also in [8], it seems appropriate to comment on how this concept compares with kniceness. That it is a strictly weaker notion is clear since 0 is always a quasi-sequentially nice submodule of G, but is not knice unless G is a fc-module.…”

Section: Warfield Modulesmentioning

confidence: 99%

“…We follow [8] in calling a p-local group G a Warfield module provided G contains a decomposition basis X such that (X) is a nice submodule and G/(X) is a totally projective p-group. Since Warfield modules are also classified in terms of their Ulm-Kaplansky invariants and Warfield invariants, it should come as no great surprise that our Axiom 3 modules are precisely the Warfield modules.…”

Section: Warfield Modulesmentioning

confidence: 99%

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Axiom 3 modules

Hill¹,

Megibben²

1986

Trans. Amer. Math. Soc.

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ABSTRACT. By introducing the concept of a knice submodule, a refinement of the notion of nice subgroup, we are able to formulate a version of the third axiom of countability appropriate to the study of p-local mixed groups in the spirit of the well-known characterization of totally projective p-groups. Our Axiom 3 modules, in fact, form a class of Zp-modules, encompassing the totally projectives in the torsion case, for which we prove a uniqueness theorem and establish closure under direct summands. Indeed Axiom 3 modules turn out to be precisely the previously classified Warfield modules. But with the added power of the third axiom of countability characterization, we derive numerous new results, including the resolution of a long-standing problem of Warfield and theorems in the vein of familiar criteria due to Kulikov and Pontryagin.

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Section: Warfield Modulesmentioning

confidence: 99%

Section: Warfield Modulesmentioning

confidence: 99%

Section: Warfield Modulesmentioning

confidence: 99%

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Axiom 3 modules

Hill¹,

Megibben²

1986

Trans. Amer. Math. Soc.

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“…We may therefore assume G is reduced. By [5,Theorem 45], there is a decomposition G = A B in which A is balanced projective (hence simply presented) and tB is simply presented. Because H is fully invariant in G we obtain a decomposition H = A B , where A and B are fully invariant subgroups of A and B, respectively.…”

Section: Proof Of Theoremmentioning

confidence: 99%

The fully invariant subgroups of local Warfield groups

Files¹

1997

Proc. Amer. Math. Soc.

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“…This will be true, for example, in the constructible universe. We first review the treatment of Ulm invariants contained in [5]. If Z is a valuated group and a is an ordinal, let kz(a) be the kernel of the map Z(a)/Z(a+ 1) ^ Z(a+ \)/Z(a + 2), so fz(a) is the dimension of kz(a) as a…”

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confidence: 99%

An application of set theory to the torsion product of abelian groups

Keef¹

1993

Proc. Amer. Math. Soc.

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Abstract. The following problem of Fuchs is considered: relate the abelian groups A and B assuming Tor(^4, G) = Tor(fl, G) for all reduced abelian groups G . A complete characterization is obtained in any set-theoretic universe in which E(k) is valid for a proper class of regular cardinals k .In the well-known list of 100 questions posed in [3], problem 50 asks us to relate the abelian groups A and B assuming Tor(^4, G) = Tor(P, G) for all reduced abelian groups G. There is clearly no loss of generality in assuming A , B, and G are p-groups for some fixed prime p, so hereafter the term "group" will mean "abelian p-group." Hill showed in [4] that A and B share many properties but left unresolved whether they must necessarily be isomorphic. Cutler showed in [1] that this need not be the case. Specifically, groups A and C were found such that C is a direct sum of cyclics, Tor(^4, G) = Tot(A © C, G) for all reduced G, but A ^ A ® C. Is this the only way to construct such examples? In other words, if A and B satisfy problem 50, do they necessarily differ by summands which are direct sums of cyclics?Denote the Ulm function of the group Y by fy . We will say A and B are T-equivalent if (a) fA(ri) = /b(«) for all n < co, and (b) for some direct sums of cyclics X and Y, A® X = B @Y.We show that if A and B are P-equivalent, then they satisfy problem 50 (Theorem 2). Conversely, if A and B satisfy problem 50 and there is a regular cardinal k greater than the ranks of A and B such the E(k) is valid, we show that A and B must be P-equivalent (Theorem 4; we review these set-theoretic terms later). In particular, we have a solution for problem 50 in any set-theoretic universe in which E(k) is valid for a proper class of regular cardinals. This will be true, for example, in the constructible universe. We first review the treatment of Ulm invariants contained in [5]. If Z is a valuated group and a is an ordinal, let kz(a) be the kernel of the map Z(a)/Z(a+ 1) ^ Z(a+ \)/Z(a + 2), so fz(a) is the dimension of kz(a) as a

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Warfield modules

Cited by 28 publications

References 8 publications

Axiom 3 modules

Axiom 3 modules

The fully invariant subgroups of local Warfield groups

An application of set theory to the torsion product of abelian groups

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