Abelian Groups - László Fuchs [Springer]

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3 Endomorphism Rings of Torsion-Free <strong>Groups</strong> 629 3 Endomorphism Rings of Torsion-Free <strong>Groups</strong> In contrast to torsion groups, non-isomorphic torsion-free groups may very well have isomorphic endomorphism rings. Another major difference in the behavior of endomorphism rings between torsion and torsion-free groups lies in the fact that only minor restrictive conditions hold for the torsion-free case. We will see in Theorem 7.1 that only slight restriction on the ring (to be cotorsion-free) is enough to guarantee that it is an endomorphism ring of a torsion-free group. Example 3.1 (Corner [2]). A ring whose additive group is isomorphic to Q ˚ Q cannot be the endomorphism ring of an abelian group. For, such a group must be torsion-free divisible, and it cannot be of rank 1, neither of rank 2, because then the rank of its endomorphism ring is 1, and 4, respectively. (Similar argument holds for Q ˚ Q ˚ Q.) Example 3.2 (Sa¸siada [4], Corner [2]). There exist torsion-free groups of finite rank whose endomorphism groups are isomorphic, but their endomorphism rings are not. Indeed, by Theorem 3.3 below, there exist such groups with endomorphism rings isomorphic to the ring Z ˚ Z and to the ring of the Gaussian integers Z C Zi. Corner’s Theorem We enter our study of torsion-free endomorphism rings with the following striking theorem. Though it is a corollary to Theorem 7.1 that is by far more general, we offer a full proof of this historically important result that opened new prospects in the theory; this proof is needed, because we will give no detailed proof for Theorem 7.1. The idea of localization in the proof is due to Orsatti [2], so Corner’s method will be needed merely in the more tractable local case. Theorem 3.3 (Corner [2]). Every countable reduced torsion-free ring is isomorphic to the endomorphism ring of some countable reduced torsion-free group. Proof. Let R be a countable, p-local reduced torsion-free ring. It is a Z .p/ -algebra, Hausdorff in its p-adic topology, so it is a pure subring in its p-adic completion QR, which is a torsion-free J p -algebra with the same identity. Choose a maximal set fz n .n
630 16 Endomorphism Rings (b) For each non-zero a 2 R choose a pair a ; a of p-adic integers such that the set f a ; a j 0 ¤ a 2 Rg is algebraically independent over S. This is possible, for S is countable, and the transcendence degree of J p over S is the continuum. Set and define the group A as the pure subgroup e a D a 1 C a a 2 Q R; (16.3) A DhR; Re a 8a 2 Ri Q R: (16.4) Obviously, A is countable, reduced, and torsion-free. (c) It is evident from the definition that A is a left R-module. It is faithful, as different elements of R act differently on 1 2 A. Consequently, R is isomorphic to a subring of End A. (d) In order to show that it is not a proper subring, select an 2 End A. SinceR is pure and dense in A (which is pure and dense in Q R), it follows that QA D Q R.By Proposition 2.10 in Chapter 6, extends uniquely to a J p -endomorphism Q of QR.Then e a DQ. a 1 C a a/ D a . Q1/ C a . Qa/ D a .1/ C a .a/ for any a 2 A. More explicitly, write p k .e a / D b 0 C nX b i e ai ; p k .1/ D c 0 C iD1 nX c i e ai ; p k .a/ D d 0 C for a i ; b i ; c i ; d i 2 R, and for some k; n 2 N. Substitution yields b 0 C nX b i . ai 1 C ai a i / iD1 D a Œc 0 C iD1 nX c i . ai 1 C ai a i / C a "d 0 C iD1 nX d i e ai iD1 # nX d i . ai 1 C ai a i / where we may assume that a 1 D a. Comparing the corresponding coefficients on both sides, we use algebraic independence to argue that b 1 D c 0 , b 1 a D d 0 , while all other b i ; c i ; d i vanish. This means p k .1/ D c 0 ; p k .a/ D c 0 a which thus holds for all a 2 R. Therefore, with the notation 1 D c 2 R, wehave a D ca for all a 2 R, showing that acts on R as left multiplication by c 2 R. The same holds for Q and for DQ A. This completes the proof of the local case. iD1
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Springer Monographs in Mathematics
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László Fuchs Abelian Groups 123
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Dedicated with love to my wife Shul
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viii Preface to relevant publicatio
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x Preface on abelian groups. The re
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xii Contents 4 Divisibility and Inj
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xiv Contents 7 Whitehead Groups If
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Table of Notations Set Theory ; : s
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Table of Notations xix N .p/: Nunk
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Table of Notations xxi Q A;;A : di
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2 1 Fundamentals is finite, countab
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4 1 Fundamentals Theorem 1.2. In a
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6 1 Fundamentals (8) A superfluous
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8 1 Fundamentals A ! A=K acting as
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10 1 Fundamentals φ G ⏐ ↓ η 0
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12 1 Fundamentals Lemma 2.5 (Snake
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14 1 Fundamentals Exercises (1) Ass
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16 1 Fundamentals non-zero subgroup
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18 1 Fundamentals in Sect. 7 in Cha
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20 1 Fundamentals 4 Sets Since the
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22 1 Fundamentals It might be helpf
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24 1 Fundamentals This is an amazin
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26 1 Fundamentals of subgroups, ind
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28 1 Fundamentals If is an infinit
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30 1 Fundamentals For the proof of
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32 1 Fundamentals C3. Identity. For
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34 1 Fundamentals is exact in D; F
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36 1 Fundamentals 7 Linear Topologi
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38 1 Fundamentals Proof. (We need s
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40 1 Fundamentals For two R-modules
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Chapter 2 Direct Sums and Direct Pr
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1 Direct Sums and Direct Products 4
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2 Direct Summands 51 Proof. If A D
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2 Direct Summands 53 Exercises (1)
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3 Pull-Back and Push-Out Diagrams 5
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4 Direct Limits 57 In this case, A
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4 Direct Limits 59 We now move to t
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5 Inverse Limits 61 for 0 W G ! A
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5 Inverse Limits 63 for every i 2 I
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6 Direct Products vs. Direct Sums 6
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6 Direct Products vs. Direct Sums 6
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7 Completeness in Linear Topologies
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Chapter 3 Direct Sums of Cyclic Gro
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1 Freeness and Projectivity 77 Coro
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1 Freeness and Projectivity 79 Let
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2 Finite and Finitely Generated Gro
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3 Factorization of Finite Groups 87
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3 Factorization of Finite Groups 89
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4 Linear Independence and Rank 91 E
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4 Linear Independence and Rank 93 t
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5 Direct Sums of Cyclic Groups 95 P
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5 Direct Sums of Cyclic Groups 97 s
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6 Equivalent Presentations 99 (5) (
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6 Equivalent Presentations 101 Thes
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6 Equivalent Presentations 103 Equi
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7 Chains of Free Groups 105 For Ded
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7 Chains of Free Groups 107 We can
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7 Chains of Free Groups 109 Hill’
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7 Chains of Free Groups 111 which i
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8 Almost Free Groups 113 (D) Let 0
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8 Almost Free Groups 115 Proof. (a)
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8 Almost Free Groups 117 be done by
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8 Almost Free Groups 119 with. In t
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9 Shelah’s Singular Compactness T
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10 Groups with Discrete Norm 123 Di
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10 Groups with Discrete Norm 125 Ch
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11 Quasi-Projectivity 127 11 Quasi-
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11 Quasi-Projectivity 129 for diffe
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132 4 Divisibility and Injectivity
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150 5 Purity and Basic Subgroups or
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152 5 Purity and Basic Subgroups Pr
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154 5 Purity and Basic Subgroups ge
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158 5 Purity and Basic Subgroups Th
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160 5 Purity and Basic Subgroups ˛
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166 5 Purity and Basic Subgroups 5
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168 5 Purity and Basic Subgroups Le
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172 5 Purity and Basic Subgroups F
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174 5 Purity and Basic Subgroups (H
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176 5 Purity and Basic Subgroups Ma
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180 5 Purity and Basic Subgroups So
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Chapter 6 Algebraically Compact Gro
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1 Algebraic Compactness 185 of the
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1 Algebraic Compactness 187 Na j ;
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1 Algebraic Compactness 189 Here jG
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2 Complete Groups 191 is a Cauchy s
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2 Complete Groups 193 A ⏐ μ A A
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3 The Structure of Algebraically Co
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3 The Structure of Algebraically Co
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4 Pure-Injective Hulls 199 (2) The
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4 Pure-Injective Hulls 201 subgroup
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5 Locally Compact Groups 203 5 Loca
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5 Locally Compact Groups 205 Theore
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6 The Exchange Property 207 Evident
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6 The Exchange Property 209 In the
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6 The Exchange Property 211 to the
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Chapter 7 Homomorphism Groups Abstr
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1 Groups of Homomorphisms 215 (and
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1 Groups of Homomorphisms 217 is an
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1 Groups of Homomorphisms 219 F Not
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2 Algebraically Compact Homomorphis
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2 Algebraically Compact Homomorphis
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3 Small Homomorphisms 225 (3) The a
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3 Small Homomorphisms 227 completio
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Chapter 8 Tensor and Torsion Produc
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1 The Tensor Product 231 with the f
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1 The Tensor Product 233 for matchi
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1 The Tensor Product 235 finitely P
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2 The Torsion Product 237 (6) (a) I
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2 The Torsion Product 239 It is pre
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2 The Torsion Product 241 It remain
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3 Theorems on Tensor Products 243 T
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4 Theorems on Torsion Products 245
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5 Localization 251 (8) (Keef) If A
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5 Localization 253 Exercises (1) If
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256 9 Groups of Extensions and Coto
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300 10 Torsion Groups (A) If C is a
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302 10 Torsion Groups the indicator
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304 10 Torsion Groups 0 −−−
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306 10 Torsion Groups is tantamount
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308 10 Torsion Groups Proof. We hav
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310 10 Torsion Groups (3) Let A be
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312 10 Torsion Groups (C) B is a ba
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314 10 Torsion Groups pure-injectiv
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316 10 Torsion Groups Theorem 3.11
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318 10 Torsion Groups 4 More on Tor
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320 10 Torsion Groups (g) The large
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322 10 Torsion Groups that in torsi
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324 10 Torsion Groups subgroup of B
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326 10 Torsion Groups (c) Homomorph
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328 10 Torsion Groups then Ker con
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330 10 Torsion Groups A class more
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332 10 Torsion Groups with basic su
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334 10 Torsion Groups 8 Valuated Ve
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336 10 Torsion Groups V D A ` B for
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338 10 Torsion Groups Proof. Define
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340 10 Torsion Groups When Socles D
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342 10 Torsion Groups Hill [1] gave
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344 11 p-Groups with Elements of In
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410 12 Torsion-Free Groups is calle
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412 12 Torsion-Free Groups Proof. L
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414 12 Torsion-Free Groups If A is
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416 12 Torsion-Free Groups Exercise
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418 12 Torsion-Free Groups Converse
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420 12 Torsion-Free Groups fa X j a
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422 12 Torsion-Free Groups and 0 !
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424 12 Torsion-Free Groups (ii) If
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426 12 Torsion-Free Groups number o
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428 12 Torsion-Free Groups along B
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430 12 Torsion-Free Groups whenever
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432 12 Torsion-Free Groups If we ha
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434 12 Torsion-Free Groups Then we
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436 12 Torsion-Free Groups (ii) For
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438 12 Torsion-Free Groups (7) (de
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440 12 Torsion-Free Groups it is ro
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442 12 Torsion-Free Groups 1 for `
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444 12 Torsion-Free Groups A D B i
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446 12 Torsion-Free Groups (5) For
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448 12 Torsion-Free Groups Proof. L
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450 12 Torsion-Free Groups local (T
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452 12 Torsion-Free Groups stable r
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454 12 Torsion-Free Groups (3) (War
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456 12 Torsion-Free Groups write x
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458 12 Torsion-Free Groups For thos
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460 12 Torsion-Free Groups A C 1
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462 12 Torsion-Free Groups ˛./.˛
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464 12 Torsion-Free Groups so the s
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466 12 Torsion-Free Groups followed
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468 12 Torsion-Free Groups (i) ther
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470 12 Torsion-Free Groups 11 Duali
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472 12 Torsion-Free Groups Lemma 11
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474 12 Torsion-Free Groups rk p D.A
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476 12 Torsion-Free Groups is proj(
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478 12 Torsion-Free Groups division
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Chapter 13 Torsion-Free Groups of I
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1 Direct Decompositions of Infinite
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2 Slender Groups 489 (6) A torsion-
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2 Slender Groups 491 Example 2.5. T
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2 Slender Groups 493 Proof. From Th
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2 Slender Groups 495 F Notes. The r
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3 Characterizations of Slender Grou
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3 Characterizations of Slender Grou
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4 Separable Groups 501 Exercises (1
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4 Separable Groups 503 where the G
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4 Separable Groups 505 In the follo
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4 Separable Groups 507 c C g D h 2
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5 Vector Groups 509 (7) Show that t
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5 Vector Groups 511 Isomorphism of
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6 Powers ofZ of Measurable Cardinal
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7 Whitehead Groups If V = L 519 (7)
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7 Whitehead Groups If V = L 521 Pro
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7 Whitehead Groups If V = L 523 Sˇ
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8 Whitehead Groups Under Martin’s
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8 Whitehead Groups Under Martin’s
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Chapter 14 Butler Groups Abstract T
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1 Finite Rank Butler Groups 531 and
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1 Finite Rank Butler Groups 533 Pro
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1 Finite Rank Butler Groups 535 Let
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2 Prebalanced and Decent Subgroups
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2 Prebalanced and Decent Subgroups
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3 The Torsion Extension Property 54
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4 Countable Butler Groups 547 (ii)
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5 B 1 -andB 2 -Groups 549 (F) Homog
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6 Solid Subgroups 551 understand th
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6 Solid Subgroups 553 fa n j n
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6 Solid Subgroups 555 Again changin
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7 Solid Chains 557 (4) A corank 1 s
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7 Solid Chains 559 B C1 =B admits
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7 Solid Chains 561 Lemma 7.6 (Bican
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8 Butler Groups of Uncountable Rank
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9 More on Infinite Butler Groups 56
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9 More on Infinite Butler Groups 57
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Chapter 15 Mixed Groups Abstract Mi
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1 Splitting Mixed Groups 575 For ev
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1 Splitting Mixed Groups 577 so A m
Page 596 and 597: 2 Baer Groups are Free 579 (4) (Opp
Page 598 and 599: 2 Baer Groups are Free 581 We note
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Page 602 and 603: 3 Valuated Groups. Height-Matrices
Page 608 and 609: 4 Nice, Isotype, and Balanced Subgr
Page 610 and 611: 5 Mixed Groups of Torsion-Free Rank
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Page 614 and 615: 6 Simply Presented Mixed Groups 597
Page 616 and 617: 7 Warfield Groups 599 Exercises (1)
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Page 620 and 621: 7 Warfield Groups 603 Lemma 7.7 (St
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Page 626 and 627: 9 Projective Properties of Warfield
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Page 630 and 631: Chapter 16 Endomorphism Rings Abstr
Page 632 and 633: 1 Endomorphism Rings 615 Lemma 1.5.
Page 634 and 635: 1 Endomorphism Rings 617 For a subg
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Page 654 and 655: 5 Special Endomorphism Rings 637 se
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Page 672 and 673: 656 17 Automorphism Groups Example
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Page 688 and 689: Chapter 18 Groups in Rings and in F
Page 690 and 691: 1 Additive Groups of Rings and Modu
Page 692 and 693: 2 Rings on Groups 677 (11) For a ri
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2 Rings on Groups 681 Beaumont-Lawv
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3 Additive Groups of Noetherian Rin
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4 Additive Groups of Artinian Rings
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4 Additive Groups of Artinian Rings
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5 Additive Groups of Regular Rings
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5 Additive Groups of Regular Rings
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6 E-Rings 693 (2) A group is isomor
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6 E-Rings 695 (F) Every endomorphis
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7 Groups of Units in Commutative Ri
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7 Groups of Units in Commutative Ri
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8 Multiplicative Groups of Fields 7
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References Books D.M. Arnold — [A
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References 709 D.M. Arnold, R. Hunt
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References 711 R. Burkhardt — [1]
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References 713 M. Dugas — [1] Fas
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References 715 A.A. Fomin — [1] T
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References 717 R. Göbel, S. Shelah
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References 719 P. Hill, M. Lane, C.
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References 721 A. Kertész — [1]
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References 723 A. Mader — [1] On
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References 725 L.G. Nongxa — [1]
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References 727 endomorphism rings.
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References 729 G.M. Tsukerman — [
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Author Index A Abel, N.H., vii Albr
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Author Index 733 Grinshpon, S.Ya.,
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Author Index 735 O’Neill, J.D., 3
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Subject Index A Abelian group, 1 Ab
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Subject Index 739 homomorphism, 37,
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Subject Index 741 Full rational gro
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Subject Index 743 Łoś-Eda theorem
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Subject Index 745 Rank 1 torsion-fr
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Subject Index 747 vector space, 334
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Abelian Groups - László Fuchs [Springer]

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