Finite torsion-free rank endomorphism rings

AbstractThe relation between the structure of a ring and the structure of its additive group is studied in the context of some recent results in additive groups of mixed rings. Namely, the notion of the square subgroup of an abelian group, which is a generalization of the concept of nil-group, is considered mainly for mixed non-splitting abelian groups which are the additive groups only of rings whose all subrings are ideals. A non-trivial construction of such a group of finite torsion-free rank no less than two, for which the quotient group modulo the square subgroup is not a nil-group, is given. In particular, a new class of abelian group for which an old problem posed by Stratton and Webb has a negative solution, is indicated. A new, far from obvious, application of rings in which the relation of being an ideal is transitive, is obtained.

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Commutative Endomorphism Rings

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-007-x ◽

1971 ◽

Vol 23 (1) ◽

pp. 69-76 ◽

Cited By ~ 6

Author(s):

J. Zelmanowitz

Keyword(s):

Abelian Group ◽

Cyclic Group ◽

Endomorphism Ring ◽

Endomorphism Rings ◽

Prime Rings ◽

Infinite Cyclic Group ◽

Free Abelian Groups ◽

Scalar Action ◽

Torsion Free

The problem of classifying the torsion-free abelian groups with commutative endomorphism rings appears as Fuchs’ problems in [4, Problems 46 and 47]. They are far from solved, and the obstacles to a solution appear formidable (see [4; 5]). It is, however, easy to see that the only dualizable abelian group with a commutative endomorphism ring is the infinite cyclic group. (An R-module Miscalled dualizable if HomR(M, R) ≠ 0.) Motivated by this, we study the class of prime rings R which possess a dualizable module M with a commutative endomorphism ring. A characterization of such rings is obtained in § 6, which as would be expected, places stringent restrictions on the ring and the module.Throughout we will write homomorphisms of modules on the side opposite to the scalar action. Rings will not be assumed to contain identity elements unless otherwise indicated.

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Decompositions of modules over a discrete valuation ring

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700012398 ◽

1979 ◽

Vol 27 (3) ◽

pp. 284-288 ◽

Cited By ~ 3

Author(s):

Robert O. Stanton

Keyword(s):

Direct Summand ◽

Valuation Ring ◽

Discrete Valuation Ring ◽

Torsion Module ◽

Direct Sum ◽

Rank One ◽

Free Rank ◽

Torsion Free Rank ◽

Torsion Free

AbstractLet N be a direct summand of a module which is a direct sum of modules of torsion-free rank one over a discrete valuation ring. Then there is a torsion module T such that N⊕T is also a direct sum of modules of torsion-free rank one.

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QUASI-ISOMORPHISMS AND GROUPS OF QUASI-HOMOMORPHISMS

Journal of Algebra and Its Applications ◽

10.1142/s0219498809003539 ◽

2009 ◽

Vol 08 (05) ◽

pp. 617-627

Author(s):

ULRICH ALBRECHT ◽

SIMION BREAZ

Keyword(s):

Abelian Group ◽

Dual Problem ◽

Abelian Groups ◽

Mixed Abelian Group ◽

Free Rank ◽

Mixed Groups ◽

Torsion Free Rank ◽

Torsion Free

This paper investigates to which extent a self-small mixed Abelian group G of finite torsion-free rank is determined by the groups Hom (G,C) where C is chosen from a suitable class [Formula: see text] of Abelian groups. We show that G is determined up to quasi-isomorphism if [Formula: see text] is the class of all self-small mixed groups C with r0(C) ≤ r0(G). Several related results are given, and the dual problem of orthogonal classes is investigated.

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Localization in non-noetherian group rings

Glasgow Mathematical Journal ◽

10.1017/s0017089500004122 ◽

1980 ◽

Vol 21 (1) ◽

pp. 151-163 ◽

Cited By ~ 1

Author(s):

P. F. Smith

Keyword(s):

Abelian Group ◽

Prime Ideal ◽

Noetherian Ring ◽

General Setting ◽

Group Rings ◽

Regular Local Ring ◽

Characteristic P ◽

Free Rank ◽

Torsion Free Rank ◽

Torsion Free

Let k be a field and G an Abelian group of finite torsion-free rank. Brewer, Costa and Lady [1, Theorem A] showed that if k has characteristic 0 then each Localization of the group algebra kG at a prime ideal is a regular local ring. They also showed (in the same theorem) that if k has characteristic p>0, then kG is locally Joetherian (i.e. each localization of kG at a prime ideal is a Noetherian ring) if and only if G is an extension of a finitely generated group by a torsion p′-group. The purpose of this note is to examine this theorem in a more general setting.

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ON PROJECTIVELY INERT SUBGROUPS OF COMPLETELY DECOMPOSABLE FINITE RANK GROUPS

Vestnik Tomskogo gosudarstvennogo universiteta Matematika i mekhanika ◽

10.17223/19988621/67/6 ◽

2020 ◽

pp. 63-68

Author(s):

Andrey R. CHEKHLOV ◽

◽

Olesya V. IVANETS ◽

Keyword(s):

Prime Number ◽

Finite Rank ◽

Invariant Subgroup ◽

Direct Sum ◽

Free Rank ◽

Torsion Free Rank ◽

Torsion Free

Let a group G be a finite direct sum of torsion-free rank 1 groups Gi. It is proved that every projectively inert subgroup of G is commensurate with a fully invariant subgroup if and only if all Gi are not divisible by any prime number p, and for different subgroups Gi and Gj their types are either equal or incomparable.

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Spectral synthesis on discrete Abelian groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004107000114 ◽

2007 ◽

Vol 143 (1) ◽

pp. 103-120 ◽

Cited By ~ 25

Author(s):

M. LACZKOVICH ◽

L. SZÉKELYHIDI

Keyword(s):

Abelian Group ◽

Abelian Groups ◽

Spectral Synthesis ◽

Free Rank ◽

Discrete Abelian Group ◽

Torsion Free Rank ◽

Torsion Free

AbstractWe prove that spectral synthesis holds on a discrete Abelian group G if and only if the torsion free rank of G is finite.

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GENERALIZED ENDOPRIMAL ABELIAN GROUPS

Journal of Algebra and Its Applications ◽

10.1142/s0219498806001545 ◽

2006 ◽

Vol 05 (01) ◽

pp. 1-17 ◽

Cited By ~ 5

Author(s):

U. ALBRECHT ◽

S. BREAZ ◽

W. WICKLESS

Keyword(s):

Abelian Group ◽

Simple Form ◽

Abelian Groups ◽

Free Rank ◽

Torsion Groups ◽

Mixed Groups ◽

Torsion Free Rank ◽

Torsion Free ◽

Nil Radical ◽

Natural Way

An n-ary endofunction on an abelian group G is a function f : Gn → G such that f(θg1,…,θgn) = θ f(g1,…,gn) for all endomorphisms θ of G. A group G is endoprimal if, for each natural number n, each n-ary endofunction has the following simple form: [Formula: see text] for some collection of integers {li : 1 ≤ i ≤ n}. The notion of endoprimality arises from universal algebra in a natural way and has been applied to the study of abelian groups in papers Davey and Pitkethly (97), Kaarli and Marki (99) and Göbel, Kaarli, Marki, and Wallutis (to appear). These papers make the case that the notion of endoprimality can give rise to interesting and tractable classes of abelian groups. We continue working along these lines, adapting our definition to make it more suitable for working with general classes of abelian groups. We study generalized endoprimal (ge) abelian groups. Here every n-ary endofunction is required to be of the form [Formula: see text] for some collection of central endomorphisms {λi : 1 ≤ i ≤ n} of G. (Note that such a sum is always an endofunction.) We characterize generalized endoprimal abelian groups in a number of cases, in particular for torsion groups, torsion-free finite rank groups G such that E(G) has zero nil radical, and self-small mixed groups of finite torsion-free rank.

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On the Reidemeister spectrum of an Abelian group

Forum Mathematicum ◽

10.1515/forum-2017-0184 ◽

2019 ◽

Vol 31 (1) ◽

pp. 199-214

Author(s):

Brendan Goldsmith ◽

Fatemeh Karimi ◽

Noel White

Keyword(s):

Abelian Group ◽

Quotient Group ◽

Abelian Groups ◽

Full Spectrum ◽

Direct Sums ◽

Reidemeister Number ◽

Free Rank ◽

Torsion Free Rank ◽

Torsion Free

Abstract The Reidemeister number of an automorphism ϕ of an Abelian group G is calculated by determining the cardinality of the quotient group {G/(\phi-1_{G})(G)} , and the Reidemeister spectrum of G is precisely the set of Reidemeister numbers of the automorphisms of G. In this work we determine the full spectrum of several types of group, paying particular attention to groups of torsion-free rank 1 and to direct sums and products. We show how to make use of strong realization results for Abelian groups to exhibit many groups where the Reidemeister number is infinite for all automorphisms; such groups then possess the so-called {R_{\infty}} -property. We also answer a query of Dekimpe and Gonçalves by exhibiting an Abelian 2-group which has the {R_{\infty}} -property.

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MODULAR ABELIAN GROUP ALGEBRAS

Asian-European Journal of Mathematics ◽

10.1142/s1793557110000192 ◽

2010 ◽

Vol 03 (02) ◽

pp. 275-293

Author(s):

Peter Danchev

Keyword(s):

Abelian Group ◽

Factor Group ◽

Compact Groups ◽

Arbitrary Group ◽

P Elements ◽

Rank One ◽

Free Rank ◽

Torsion Groups ◽

Torsion Free Rank ◽

Torsion Free

Suppose FG is the F-group algebra of an arbitrary multiplicative abelian group G with p-component of torsion Gp over a field F of char (F) = p ≠ 0. Our theorems state thus: The factor-group S(FG)/Gp of all normed p-units in FG modulo Gp is always totally projective, provided G is a coproduct of groups whose p-components are of countable length and F is perfect. Moreover, if G is a p-mixed coproduct of groups with torsion parts of countable length and FH ≅ FG as F-algebras, then there is a totally projective p-group T of length ≤ Ω such that H × T ≅ G × T. These are generalizations to results by Hill-Ullery (1997). As a consequence, if G is a p-splitting coproduct of groups each of which has p-component with length < Ω and FH ≅ FG are F-isomorphic, then H is p-splitting. This is an extension of a result of May (1989). Our applications are the following: Let G be p-mixed algebraically compact or p-mixed splitting with torsion-complete Gp or p-mixed of torsion-free rank one with torsion-complete Gp. Then the F-isomorphism FH ≅ FG for any group H implies H ≅ G. Moreover, letting G be a coproduct of torsion-complete p-groups or G be a coproduct of p-local algebraically compact groups, then [Formula: see text]-isomorphism [Formula: see text] for an arbitrary group H over the simple field [Formula: see text] of p-elements yields H ≅ G. These completely settle in a more general form a question raised by May (1979) for p-torsion groups and also strengthen results due to Beers-Richman-Walker (1983).

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