Infinite Abelian Groups and Homological Methods

Harrison, David K.

doi:10.2307/1970188

Cited by 170 publications

(103 citation statements)

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“…The latter were classified by Harrison [13] as products G = Y\Tp over all primes p where T is a direct summand of Zx for some X. Furthermore, G is uniquely determined by the dimensions of T/pT over Z/(p).…”

Section: In X If F Designates This Module Then T -> K(p)(x) Is a Flamentioning

confidence: 99%

Flat Covers and Flat Cotorsion Modules

Enochs¹

1984

Proceedings of the American Mathematical Society

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Abstract. It is not known whether modules over an arbitrary ring have flat covers, however for certain modules over commutative noetherian rings they can be shown to exist. These covers, in turn, have an interesting connection with flat cotorsion modules. A complete description of flat cotorsion modules analogous to that given by Harrison for torsion free, cotorsion abelian groups will be given.In this article, R will denote a commutative noetherian ring.Notation. If R is a local ring, m(R) will denote its maximal ideal. For p g Spec(Ä), Rp will denote the completion of the local ring Rp, and for any R-module, M will denote the (separated) completion of the Rp module Mp with the m(Ä )-adic topology. k(p) denotes the residue field of R (= Rp/m{Rp)). E(M) will be an injective envelope of M. If R is local, M" denotes the Maths dual,of M, and M is said to be reflexive if Mvv = M naturally.For a set X, Mx is the module of all functions X -* M and M(X) the submodule of those functions with finite support. Soc(M) will denote the socle of M.

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Section: In X If F Designates This Module Then T -> K(p)(x) Is a Flamentioning

confidence: 99%

Flat Covers and Flat Cotorsion Modules

Enochs¹

1984

Proceedings of the American Mathematical Society

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“…Let S denote the group of finite sequences in F and, for the purposes of our next lemma and theorem, let F denote the cotorsion completion of 5. (For information concerning cotorsion groups, see [9].) Lemma 4.1.…”

Section: A Group G Is a Baer Group If And Only If It Is Freementioning

confidence: 99%

A solution to the splitting mixed group problem of Baer

Griffith¹

1969

Trans. Amer. Math. Soc.

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“…It is known (see [7], [8]) that A admits a compact topology exactly if it is a direct product of groups of type Z(p n ), Z(p°°), J v and R where the number of groups Z(p°°), for any p, does not exceed that of R. All these groups admit linearly compact topologies, so by (c) the assertion follows.…”

Section: A Group Which Admits a Compact Topology Admits A Linearly Comentioning

confidence: 99%

Note on Linearly Compact Abelian Groups

Fuchs

1969

J. Aust. Math. Soc.

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1By a group A is meant throughout an additively written abelian group. A is said to have a linear topology if there is a system of subgroups U\ (i el) of A such that, for any a e A, the cosets a-\-U i (i el) form a fundamental system of neighborhoods of a. The group operations are continuous in any linear topology; the topologies are always assumed to be Hausdorff, that is, P|,-.U t = 0. A linearly compact group is a group A with a linear topology such that if a,-f^4,-(j el) is a system of cosets modulo closed subgroups Aj with the finite intersection property (i.e. any finite number of a^Aj have a non-void intersection), then the intersection |~) 3 (a^Aj) of all of them is not empty.Our present aim is to study the algebraic structure of linearly compact groups. They can be characterized as inverse limits of groups with minimum condition on subgroups. It turns out that the class of linearly compact groups lies between the classes of compact and algebraically compact groups: every group that admits a compact topology is linearly compact in some suitable topology, while all linearly compact groups are algebraically compact. A complete system of invariants can be obtained for linearly compact groups. In obtaining these invariants, a fundamental role is played by the fact that every linearly compact group A can be decomposed as a direct product A -Y^^Aj, where, for each prime p, A v is a topological module over the />-adic integers, with the product topology on A. For the components A p we can apply the duality theory of Kaplansky [10] and Schoneborn [15] (between discrete and linearly compact />-adic modules) in order to get a full structure theorem on linearly compact groups.We shall use the following notations: Z = infinite cyclic group; Z(n) = cyclic group of finite order n; Z(p°°) = group of type p°°; Q = group of rational numbers; R = group of real numbers; K = real numbers mod 1; J v = group of ^>-adic integers; K v = group of ^>-adic numbers. The symbols © and ]T denote direct sums and products, respectively.

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Infinite Abelian Groups and Homological Methods

Cited by 170 publications

References 1 publication

Flat Covers and Flat Cotorsion Modules

Flat Covers and Flat Cotorsion Modules

A solution to the splitting mixed group problem of Baer

Note on Linearly Compact Abelian Groups

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