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Forum Math., Ahead of PrintDOI 10.1515/forum-2013-0111

Forum Mathematicum© de Gruyter 2014

QE-transitive abelian groups

Manfred Dugas

Communicated by Manfred Droste

Dedicated to the memory of Jutta Hausen

Abstract. Let G be a torsion-free abelian group and n be a (finite) cardinal. The group Gis called n-QE-transitive if for any pure subgroups X; Y of rank n of G, there existssome endomorphism ' of G such that '.X/ � Y and Y='.X/ is a torsion abelian group.We investigate these groups and obtain results similar to the ones for E-transitive groups.

Keywords. QE-transitive torsion-free abelian groups, Q-separable groups.

2010 Mathematics Subject Classification. 20K20, 20K30.

1 Introduction

The word ‘group’ in this paper will always mean ‘torsion-free abelian group’ andour notations are standard as in [9] or [12]. LetG be a group. Then End.G/ denotesthe ring of endomorphisms of G and, for any cardinal n, we use Ln.G/ to denotethe set of all pure subgroups of rank n of G. Recall that the group G is calledstrongly homogeneous if Aut.G/ operates transitively on the set L1.G/, and Gis called E-transitive if for all X; Y 2 L1.G/ there exists some 2 End.G/ suchthat .X/ D Y . We will generalize these notions by introducing the following:

Definition 1.1. Let G be a group and n be a cardinal. Then G is n-QE-transitiveif for all X; Y 2 Ln.G/ there exists some 2 End.G/ such that .X/ � Y andY=. .X// is torsion, i.e. Y is the purification of .X/ inG. IfG is 1-QE-transitive,we simply call the group G QE-transitive.

Strongly homogeneous groups G of finite rank were characterized in [1] andstrongly homogeneous groups G of infinite rank were investigated by Krylovin [13]. It was shown that G Š A˝Z F , where A is some group of rank 1 and Fis an @1-free module over the strongly homogeneous ring C D Center.End.G//.Thus, if G is countable, then F is a free C -module. J. Hausen obtained the sameconclusion for E-transitive groups in [10] and extensions of this result can be foundin [11] and [6].

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2 M. Dugas

Note that “QE-transitive” means “E-transitive” in the quasi-category of torsion-free abelian groups. In Section 4 we present an example to show that QE-transitiveis indeed weaker than E-transitive. Here is a list of some of our results:

� If G is QE-transitive, then G is homogeneous, i.e. all non-zero elements of Ghave the same type.

� We present examples of groups G which are QE-transitive but not E-transitive.

� G is n-QE-transitive if and only if Q End.G/ operates transitively on Ln.QG/(Q_ is short for Q˝Z _ ).

� If G is n-QE-transitive and 1 � n < rank.G/, then G is QE-transitive.

� IfG is QE-transitive of infinite rank such thatD WD ¹ı 2 EndQ.QG/ W ı D ıfor all 2 End.G/º D Q, then G is n-QE-transitive for all n 2 N.

� Let G be a torsion-free 2-QE-transitive group of rank at least 4. Then G isn-QE-transitive for all n 2 N and D D Q. Moreover, pure, countable sub-groups of G are homogeneous completely decomposable.

� LetG be a countable torsion-free 2-QE-transitive group of rank at least 4. ThenG is homogeneous completely decomposable and thus strongly homogeneous.

� There exist 2-QE-transitive groups of rank 3 which are strongly indecompos-able but not strongly homogeneous with D D Q End.G/ of rank 3.

� Let F be a free module over a Dedekind domain S . Let F1; F2 be infinite ranksubmodules of F such that F1 is pure and rank.F1/ � rank.F2/. Then thereexists some ' 2 EndS .F / such that '.F1/ D F2 D '.F /.

� Let G be a torsion-free @0-QE-transitive group of infinite rank. Then one ofthe following holds:

(1) G is n-QE-transitive for all n 2 N and D D Q.

(2) There exists a pure, finite rank fully invariant subgroup V ¤ 0 such thatG=V belongs to case (1).

(3) G is the union of a countable chain of pure finite rank fully invariant sub-groups.

� We will give examples to show that all three cases in the previous item actuallyoccur. Note that groups with properties (2) or (3) are not QE-transitive.

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QE-transitive abelian groups 3

2 QE-transitive groups

For a groupG, letR D End.G/ denote the ring of endomorphisms ofG. Note thatif G is a group, then G is a faithful left R-module, and we set QG D Q˝Z G

and we have that the vector space QG is a faithful left module over the Q-algebraQ End.G/. The next proposition follows directly from the definitions:

Proposition 2.1. Let G be a group, n a cardinal and R D End.G/. Then the fol-lowing are equivalent:

(1) G is n-QE-transitive.

(2) QR operates transitively on the set Ln.QG/.

It is easy to see that any QE-transitive group is homogeneous of some type � :Let X; Y be pure subgroups of rank 1 of G. Then there is some ' 2 End.G/ suchthat '.X/ ¤ ¹0º is a subgroup of Y , which is the purification of '.X/ in G. It fol-lows that type.X/ � type.'.X// � type.Y /. By symmetry, type.X/ D type.Y /.

Recall that a group G is called irreducible if ¹0º and G are the only pure, fullyinvariant subgroups of G. Next we show:

Proposition 2.2. Let G be a group. Then:

(1) G is QE-transitive if and only if G is irreducible.

(2) G is QE-transitive if and only if QG is a simple QR-module.

(3) If G is n-QE-transitive and 1 � n < rank.G/, then G is QE-transitive.

Proof. To show (1), assume that G is QE-transitive and let ¹0º ¤ P be a pure,fully invariant subgroup of G. There exists some X 2 L1.G/ with X � P . Forany Y 2 L1.G/ there is some 2 R such that ¹0º ¤ .X/ � Y and Y is the pu-rification of .X/ inG. It follows that Y � P and thus P D G. On the other hand,if G is irreducible, and X 2 L1.G/, then the purification of R.X/ is a pure, fullyinvariant subgroup of G and thus equal to G. Let Y 2 L1.G/. Then there existssome 2 End.G/ such that ¹0º ¤ .X/ � Y and we have thatG is QE-transitive.

Statement (2) follows from (1). To show (3), let ¹0º ¤ P be a pure, fully invari-ant subgroup of G. Assume rank.P / � n. Then each Y 2 L1.G/ is contained ina pure subgroup C of G of rank n and P contains a pure subgroup A of rank n.There is some 2 R such that .A/ � C and C is the purification of .A/ whichis contained in P . It follows that Y � P and thus P D G and G is irreducible.By (1), G is QE-transitive. Now assume that k D rank.P / < n. Let X be a purerank n subgroup of G such that P � X . Let Y be a pure rank n subgroup of G.Then there exists some ' 2 End.G/ such that '.X/ � Y with Y='.X/ torsion.

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The map '�X is injective and '.P / � P , which implies that P is the purifica-tion of '.P / and thus P � Y . It follows that all pure, rank n subgroups of Gcontain P . This can only happen if n D rank.G/, a contradiction.

We will fix the following:

Notation 2.3. Let G be a QE-transitive group and R D End.G/. We view G asa left R-module. Let D D ¹ı 2 EndQ.QG/ W ı D ı for all 2 Rº denote thecentralizer of R in EndQ.QG/. By Proposition 2.2 (2), Schur’s lemma appliesandD is a division ring containing Q. It is natural to view QG as a rightD-vectorspace, since then QG becomes a QR-D-bimodule and QG D

Lb2B bD is a right

D-vector space with D-basis B . Moreover, all maps in QR are D-linear transfor-mations of the D-vector space QG. We may, and will, assume that B is a subsetof G. For each b 2 B , there is a subgroup Tb of D such that G \ bD D bTb .

Recall that the ring S with additive group SC is called an E-ring if for any endo-morphism ' of SC there exists some a 2 S such that '.x/ D ax for all x 2 SC.It is well known that E-rings are commutative with identity.

Proposition 2.4. LetG be QE-transitive of infinite rank such thatD D Q. ThenGis n-QE-transitive for all n 2 N. Moreover, if X; Y are pure finite rank subgroupsofG with rank.X/ � rank.Y /, then there exists some ' 2 End.G/ with '.X/ � Yand Y='.X/ torsion.

Proof. Note that QG is a simple, faithful QR-module. Let ¹ui W 1 � i � nº bea set of Q-linearly independent elements of G and ¹vi W 1 � i � nº be a set of atmost n elements of G. By the Jacobson Density Theorem [12], there exists some 2 QR such that .ui / D vi for all 1 � i � n. Thus there exists some k 2 Nsuch that k D ' 2 R and '.ui / D kvi for all 1 � i � n. The result follows.

Let b; b0 2 B . Since G is QE-transitive, there exists some ' 2 R and m 2 Nsuch that '.b/ D mb0. Then

b0mTb D '.bTb/ D '.G \ bD/

� G \ '.bD/ D G \ .'.b/D/ D G \ b0mD

� G \ b0D D b0Tb0 :

We infer that mTb � Tb0 and, by symmetry, Tb is quasi-equal to Tb0 for allelements b; b0 2 B . Set T D Tb .

We are now ready for:

Proposition 2.5. Let V be a finite dimensional subspace of .QG/D and considerW D G \ V . Then W is quasi-isomorphic to a finite direct sum of copies of T .

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QE-transitive abelian groups 5

Proof. We will induct over n D dimD.V /. If n D 1, then we may assume thatV D bD for some b 2 B and W D G \ bD is quasi-isomorphic to T . Now letV D U ˚bD andW0 D G\U . By induction hypothesis,W0 is quasi-isomorphicto a finite direct sum of copies of T . By the Jacobson Density Theorem, thereexists some � 2 QR such that �.U / D ¹0º and �.b/ D b. Thus there exists somem 2 N such that � D m� 2 R and m D � C .m � �/. It follows that

mW � �.W /C .m � �/.W /

� .G \ bD/˚ .G \ U/ D bTb ˚W0

� G \ V D W:

This shows that W is quasi-isomorphic to bT ˚W0.

Corollary 2.6. Let G be QE-transitive such that dim..QG/D/ D n is finite. ThenG is quasi-isomorphic to a direct sum of n copies of T and T is a strongly inde-composable QE-transitive group. Moreover, there exists a full rank subring S ofDsuch that S � T � QS D D and End.T / D S:

Proof. By Proposition 2.5, there exists a finite subset B of G such that G isquasi-isomorphic to M D

Lb2B bTb and bTb D G \ bD are quasi-isomorphic

for all b 2 B . Thus there exists some m 2 N such that mG �M � G.Let � 2 R such that �.b/ 2 bTb . Then �.bTb/ � G \ bD D bTb . This shows

that � 2 Rb D ¹� 2 R W �.bTb/ � bTbº. Note that Rb is a subring of R.Let � 2 End.bTb/. Then � extends to an endomorphism of M that we denote

by � again. Let g 2 G. Then �.mg/ 2M � G and thus m� 2 Rb . This showsthat mEnd.bTb/ � .Rb/�bTb

� End.bTb/.Let 0 ¤ bs; bt 2 bTb . Since G is QE-transitive, there exists some ˛ 2 R and

k 2 N such that ˛.bs/ D kbt and thus ˛.b/ 2 ˛.bD/ \G D bD \G D bTb .We infer ˛ 2 Rb and it follows that bTb is QE-transitive. Moreover, D is con-tained in the centralizer Db of End.bTb/.

Let Sb D Rb=annR.b/ Š .Rb/�bTb. Then

bSb � b End.bTb/ � bTb � bQSb D bD;

which shows that bTb is strongly indecomposable.

Remark 2.7. Zassenhaus has shown that if S is a ring with free additive group,then there exists some T satisfying the last sentence of Corollary 2.6. We referto [5] and the literature referenced there for an extension of Zassenhaus’ result tosome rings S with free additive group of countable rank.

Next we deal with the case of countable D-dimension.

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6 M. Dugas

Proposition 2.8. Let G be QE-transitive such that QG DLi2N biD has count-

able D-dimension. Let Vn DLniD1 biD and Gn D G \ Vn. Then the group Gn

is QE-transitive and a quasi-summand of GnC1 for all n. Moreover, GnC1=Gn isquasi-isomorphic to T . Note that G D

Sn�1Gn.

Proof. Note that V D QG is a simple, faithful QR-module. Let 0 ¤ g; g0 2 Gn.We may assume that ¹g; b2; : : : ; bnº is a D-basis of Vn. By the Jacobson Den-sity Theorem there is some 2 QR such that .g/ D g0 and .bi / 2 Gn. Thisshows that n D ' 2 R for some natural number n and '�Gn

W Gn ! Gn suchthat '.g/ D ng0. We infer that Gn is QE-transitive for all n � 1, and the rest fol-lows from Proposition 2.5.

Later we will construct an example of such a group G that is not quasi-isomor-phic to a direct sum of copies of T .

Let R be a torsion-free ring and let G be a group such that R � G � QR andEnd.G/ D R� . Let D D ¹˛ 2 EndQ.QR/ W ˛' D '˛ for all ' 2 End.G/º. Let˛ 2 D. Then ˛.rx/ D r˛.x/ for all r 2 R. It follows that ˛ D �.˛.1// is a multi-plication and thus D D �.QR/. If QR is a division ring, then G is QE-transitive.

Proposition 2.9. Let R be a torsion-free ring and let G be a group such thatR � G � QR and End.G/ D R� . Then G is QE-transitive if and only if QR isa division ring. Moreover, G is E-transitive if and only if G D AR, where R isan E-ring that is a strongly homogeneous principal ideal domain and A is somesubgroup of Q.

Proof. The first assertion is easy to show. Assume that the groupG is E-transitive.By [10, Proposition 2.3], we haveD DQZ.R/ andD D �.QR/ by the above. Thisshows that R is commutative. By [10, Theorem 3.4] we have that R is a stronglyhomogeneous principal ideal domain. By [10, Proposition 2.2],G D AF where Fis E-transitive and E-cyclic. By [10, Propostion 4.3], R D Z.R/ is an E-ring andF is a free R-module of rank 1, i.e. F D R. The converse follows easily from thedefinitions since each element of R is an integer multiple of a unit of R.

3 2-QE-transitive groups

Our goal in the section is to prove the following

Theorem 3.1. Let G be a torsion-free, 2-QE-transitive abelian group with theproperty that rank.G/ � 4. Then G is n-QE-transitive for all n 2 N. Moreover,if H is a pure subgroup of G with rank.H/ � @0, then H is homogeneous com-pletely decomposable.

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QE-transitive abelian groups 7

We maintain the notations of the previous section.

Proposition 3.2. LetG be a 2-QE-transitive group. Then either bD D QG for any0 ¤ b 2 G or D D Q. In other words, dimD.QG/ D 1 or else dimQ.D/ D 1.

Proof. Suppose dimD.QG/ > 1 and dimQ.D/ > 1. Pick an element d 2 D �Qand b; b0 2 G someD-linearly independent elements ofG. Then there exists some 2 R D End.G/ such that .¹b; bdº/ � hb; b0i� such that hb; b0i� = h .hb; bd i/iis a torsion group. But .bd/ D .b/d 2 .b/D and .¹b; bdº/ is contained ina 1-dimensional D-subspace of QG. A contradiction.

Before we proceed, we recall a result due to Baer [3], see also [9, Proposi-tion 86.5].

Proposition 3.3 ([3]). Let C be a pure subgroup of the torsion-free group A suchthat:

(a) A=C is completely decomposable and homogeneous of type � ,

(b) all elements in A � C are of type � .

Then C is a direct summand of A

This proposition can be applied to show that any finite extension A of a ho-mogeneous, completely decomposable group C is completely decomposable andthus A and C are isomorphic. We are now ready for:

Theorem 3.4. Let G be 2-QE-transitive. There are two cases:� R � G � D D QR, R D End.G/ and D is a field with ŒD W Q� � 3,� D D Q and each pure, countable rank subgroup ofG is completely decompos-

able and homogeneous of type � .

In the latter case, G is n-QE-transitive for all finite n < rank.G/.

Proof. First we deal with the case of QG D D. Let u; v 2 R such that h1; ui�and h1; vi� are pure rank-2 subgroups of G. Then there exists an r 2 R suchthat r h1; ui� � h1; vi� and has rank 2. It follows that r 2 Q.v/, the subfieldof D generated by v over Q and ru 2 Q.v/. We infer that u 2 Q.v/ and thusD D Q.v/ is a field. If v2 2 Q, then we are done. Now assume that ¹1; v2º and¹1; vº are linearly independent over Q. Then there exists an r 2 Q˚Qv suchthat rv2 2 Q˚Qv since G is 2-QE-transitive. It follows that the minimal poly-nomial of v over Q has degree at most 3. In the second case, we have D D Q andG \ bD D bTb for some subgroup Tb of Q, all of which have the same type � andthus are all isomorphic to each other. By Proposition 3.3, we get that each Gn is adirect summand in GnC1 and each Gn is completely decomposable of type � .

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8 M. Dugas

Corollary 3.5. Let G be a 2-QE-transitive group such that 4 � rank.G/ � @0.Then G is homogeneous completely decomposable.

Corollary 3.6. Let G be a 2-QE-transitive group of rank at least 4. Let n 2 N and¹ui W 1� i � nº be a linearly independent set of elements ofG. Let ¹vi W 1� i � nºbe a set of elements ofG. Then there exists some ' 2 End.G/ and 0 ¤ m 2 Z suchthat '.ui / D mvi for all 1 � i � n.

Proof. By Theorem 3.4, we have that G is QE-transitive and D D Q. As in theproof of Proposition 2.4, we may apply Jacobson’s density theorem and find some 2 QR such that .ui / D vi for all 1 � i � n. There exists some m 2 N suchthat ' D m 2 R has the desired property.

Let G be a homogeneous group of type � D .�p/p2P . Let

P D ¹p 2 P W �p D1º and Q D ¹p 2 P W 1 � �p <1º:

Let S D ZŒ 1pW p 2 P �, a subring of Q of type � , and A D

Pp2Q p

��p Z. ThenAS , which is short for A˝Z S , has type � . Define L.G/ D

Tp2Q p

�pG.

Claim 3.7. We have G D AL.G/ and L.G/ is a S -module of type � .

Proof. Clearly, AL.G/ � G. Let 0 ¤ g 2 G. Let Q0 D ¹p 2 Q W hp.g/ D �pº,Q1 D ¹p 2 Q W hp.g/ > �pº,Q2 D ¹p 2 Q W hp.g/ < �pº. ThenQ1 andQ2 arefinite sets. Let m D

Qp2Q2

p�p�ep , where ep D hp.g/. Then

mg 2 L.G/ and1

mD

Yp2Q2

pep

p�p2 A:

It follows that g D 1m.mg/ 2 AL.G/ and thus G D AL.G/.

Note thatH D L.G/ is a fully invariant subgroup ofG and since Z D End.A/,it is easy to see that End.G/ D End.H/.

Corollary 3.8. Let G be a 2-QE-transitive group with rank.G/ � 4. Then one hasG D AH , where A is a rank-1 group and H is a 2-QE-transitive group and an@1-free S -module for some subring S of Q. Moreover, H is 2-QE-transitive withEnd.H/ D End.G/ and S is the center of End.G/.

Proposition 3.9. Let F be a field extension of Q with dimQ.F / D 3. Then F �operates transitively on L2.F /.

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QE-transitive abelian groups 9

Proof. Pick X; Y 2 L2.F /. We want to find an element t 2 F such that tX D Y .We may assume that X D 1Q˚ vQ and Y D 1Q˚ uQ. There exists u0 2 Ywith minimal polynomialmu0.x/ D x3 C ax C b 2 QŒx�. (See [12, proof of Pro-position 4.8].) We may replace u by u0 without changing Y . Note that F D QŒu�and we may write v D ˛u2 C ˇuC for some rational numbers ˛; ˇ; with˛ ¤ 0 since otherwise X D Y and there is nothing to show in this case.

Define t D �ˇ C ˛u 2 Y . We compute

tv D .�ˇ C ˛u/.˛u2 C ˇuC /

D u20C u.�ˇ2 � a˛2 C ˛ / � ˇ � ˛2b 2 Y:

This shows that tX D Y .

Corollary 3.10. Let F be a field extension of Q with dimQ.F / D 3 and let G bea subgroup of the additive group of F such that QG D F and Q.End.G// � F �.Then G is 2-QE-transitive.

Corollary 3.11. Let F be a field extension of Q with dimQ.F / D 3 and let R bea subring of F with QR D F . If G � F has rank 3 and is an R-module, then Gis 2-QE-transitive.

Recall that the subgroupB is a quasi-summand ofA if there exists somem 2 Nsuch that, for some subgroup C of A, we have mA � B ˚ C � A. Note that if Bis a quasi-summand of A, then B�, the purification of B in A, is also a quasi-summand of A. We call the group A 1-Q-separable if each pure rank-1 subgroupof A is a quasi-summand of A. We have:

Proposition 3.12. LetB be a pure subgroup of the 1-Q-separable groupA. ThenBis 1-Q-separable.

Proof. Let X be a pure rank-1 subgroup of B . Then X is also a pure rank-1 sub-group of A and thus there exists some m 2 N and some subgroup C � A suchthat mA � X ˚ C � A. Intersecting this with B yields

mB D mA \ B � .X ˚ C/ \ B D X ˚ .C \ B/ � A \ B D B:

This shows that B is 1-Q-separable.

We call the group A Q-separable if each pure, finite rank subgroup of A isa quasi-summand.

Proposition 3.13. Any 1-Q-separable group A is also Q-separable.

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10 M. Dugas

Proof. Let B be a pure subgroup of A of rank n. We induct over n. If n D 1,there is nothing to show because A is 1-Q-separable. Assume that B has rank nand let B 0 be a pure subgroup of B of rank n � 1. By induction hypothesis, thereexists some m 2 N and a pure subgroup C � A such that mA � B 0 ˚ C � A.There exists some b 2 B such that B D hB 0; bi�. We have that mb D b0 C c forsome b0 2 B 0 and c 2 C \ B . By Proposition 3.12, we have that hci� is a quasi-summand of C , i.e. kC � hci� ˚ C

0 � C . It follows that

mkA � kB 0 ˚ kC � kB 0 ˚ .hci� ˚ C0/ � B 0 ˚ hci� ˚ C

0� A:

This shows that B 0 ˚ hci� is a quasi-summand of A and so is B , its purificationin A.

Corollary 3.14. Let A be a 1-Q-separable group that is homogeneous of type � .Then each pure finite rank subgroup B is a quasi-summand of A and B is com-pletely decomposable and homogeneous of type � . If C is a pure subgroup of A ofcountable rank, then C is completely decomposable.

Proof. The proof of the previous proposition shows that B is a quasi-summandof A and mkB � B 0 ˚ hci� � B . By induction hypothesis, B 0 is completely de-composable of type � . That B is completely decomposable is an immediate con-sequence of [2, Theorem 2.3]. Moreover, an inspection of the proof of this resultshows that B 0 is a direct summand of B . Thus C is the union of a chain of finiterank subgroups Bn such that Bn is a direct summand of BnC1 for all n and itfollows that C is completely decomposable.

Theorem 3.15. Let G be a 2-QE-transitive group of rank at least 4 such that thereexists some 2 End.G/ such that .G/ is countable. Then G is Q-separable.

Proof. Let G D H ˝ L.G/, where H is a subgroup of Q where all elementshave finite p-height for all primes p and L.G/ is an S -module for some sub-ring S of Q such that G is homogeneous of type type.H ˝ S/. Then L.G/ is2-QE-transitive and an @1-free S -module. This shows that .L.G// is containedin some free S -module and is thus free because S is a principal ideal domain. It fol-lows that G has a quasi-direct summand U of rank 1, i.e. mG � U ˚G0 � G forsuitable m;G0. Let W be a pure rank-1 subgroup of G. Since G is also 1-QE-transitive, there exists some 2 End.G/ such that kU � .W / � U for somenatural number k. Let � be the endomorphism of U ˚G0 such that �.G0/ D ¹0ºand ��U D idU . Then � induces a map from mG to G. Let g 2 W \ ker. �/.Then 0 D .g /� 2 U� . It follows that g D 0 and thus g D 0 because �W isinjective. We have thatW \ker. �/ D ¹0º. Note that kU D kU� � W � G and thus mkU D mkU� � mW � � mG � � U . This shows that there exist

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QE-transitive abelian groups 11

natural numbers a; b that are multiples of m such that aW � D bG � and thusfor all g 2 G there is some w 2 W with bg � aw 2 G \ ker. �/. It follows thatbG � W ˚ .G \ ker. �// � G and we infer that W is a quasi-summand of Gand G is 1-Q-separable. By Proposition 3.13, G is Q-separable.

Corollary 3.16. Let G be a countable 2-QE-transitive group of rank at least 4.Then G is homogeneous completely decomposable.

4 An example

Assume that S is an E-ring with torsion-free additive group SC. It is easy to seethat SC is QE-transitive if and only if QS is a field. On the other hand, assumethat R is a strongly homogeneous E-ring, i.e. each element in R in an integermultiple of a unit of R. Then RC is strongly homogeneous. The ring Jp of p-adicintegers is an example of such a ring.

We will construct a class of groups to illustrate the difference of the notions ofE-transitive and QE-transitive.

Let S be a ring such that the additive group of S is torsion-free and p-reducedfor some fixed prime number p. Let B D ¹f º [ ¹fi W i D 0; 1; 2; : : : º be a basisof a free right S -module F of countable rank. Pick a p-adic number

� D

1X˛D0

p˛�˛

such that �0 ¤ 0 and 0 � �˛ < p for all ˛. Let

ai D

i�1X˛D0

p˛�˛ and xi D p�i

fai �

i�1X˛D0

p˛f˛

!2 QF :

Now we define

H D H.�; S/ D F C

1XiD1

xiZ:

It is easy to see:

Observation 4.1. One has H=F Š Z.p1/, which implies that H and F are notquasi-equal.

Observation 4.2. LetX be a finite subset ofB , b 2 B �X , and v 2 F . Then thereexists some S -linear map ' 2 End.FS / and some m 2 N such that '.b/ D vpm,'.X/ D ¹0º and '.H/ � F , i.e. ' induces an endomorphism of H .

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12 M. Dugas

First we consider the case b D f . We may assume thatX D ¹f0; f1; : : : ; fk�1º.There exists 2 End.FS / such that .f / D vpk , .fi / D 0 for 0 � i � k � 1and .f˛/ D v�˛�k for all ˛ � k. Now we compute for i > k

pi .xi / D vpkai �

i�1X˛Dk

vp˛�˛�k D v

i�1X˛D0

p˛Ck�˛ �

i�1�kX˛D0

p˛Ck�˛

!

D v

i�1X˛Di�k

p˛Ck D v

iCk�1X˛Di

p˛ 2 piF:

This shows that .xi / 2 F for all i > k. Now let ' D pk and m D 2k.Let now b D fi for some index i � 0. Pick some k 2 N such that i < k and

X � ¹f º [ ¹f˛ W 0 � ˛ � k � 1; ˛ ¤ iº. Again we define some ' 2 End.FS /by '.fi / D vpk , '.fk/ D �vpi and '.f / D 0 D '.f˛º for all ˛ … ¹i; kº. Forj � k we compute

pj'.xj / D �.pi'.fi /C p

k'.fk// D �.vpiCk� vpkCi / D 0:

This shows that D pk' has the desired properties.

Observation 4.3. Assume that QS is a division ring. Then H is QE-transitive.

Let 0 ¤ u 2 F and u D bt C x, where t ¤ 0, b 2 B and x 2Lb¤c2B cS .

By Observation 4.2 there exists some 2 End.H/ \ End.FS / such that .x/ D 0and .b/ D bspk for some k 2 N where st D n 2 N. Such an element s existsbecause QS is a division ring. Let v 2 F . There exists ı 2 End.H/ \ End.FS /with ı.b/ D vp` for some ` 2 N. We have

.ı ı /.u/ D ı. .bt// D ı. .b/t/ D ı.bspkt / D ı.bnpk/ D vnpkC` 2 vZ:

This shows that H is QE-transitive.Naturally, F DS FS is a S -bimodule. Let

D D ¹˛ 2 End.QF / W ˛' D '˛ for all ' 2 End.H/º:

Observation 4.4. We have D � �.QS/.

Let ˛ 2 D and b 2 B with the property that ˛.b/ D btb C x. Then there existssome ' 2 End.H/ \ End.FS / with '.x/ D 0 and '.b/ D bpk for some k 2 N.Then

btbpkD '.btb C x/ D ˛.bp

k/ D btbpkC xpk

and it follows that˛.b/ D btb 2 bS D Sb:

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QE-transitive abelian groups 13

Moreover, there exists some 2 End.H/ \ End.FS / such that .b/ D cpk and .c/ D bpk for any b; c 2 B . It follows that

ctbpkD .btb/ D .˛.b// D ˛. .b// D ˛.cp

k/ D ctcpk

and thus tb D t for all b 2 B . Let r 2 S . There exists some � 2 End.H/\End.FS /with �.b/ D brpk . Then brpkt D �.bt/ D �.˛.b// D ˛.�.b// D ˛.brpk/ andthus ˛.rb/ D ˛.br/ D brt D r.bt/ D r˛.b/ for all b 2 B and r 2 S . Now it iseasy to see that ˛ 2 End.S .QF // is a S -linear map of the left S -module and˛ D �t 2 �.QS/.

We may consider Observation 4.2 by letting all ' be in End.H/ \ End.SF /.Then we get:

Observation 4.5. One has D � QZ.S/, where Z.S/ denotes the center of thering S .

There exists some 2 End.H/ \ End.SF / such that .b/ D spkb. Maintain-ing the above notation, we have

spktb D ˛.spkb/ D ˛. .b// D .˛.b// D .tb/ D t .b/ D tspkb:

It follows that ts D st for all s 2 S .

Observation 4.6. If EndZ.S/.S/ D EndZ.S/, then one has D D Q.Z.S// andZ.End.H// D S0, where S0 is the p-adic closure of Z in S . Note that S0 � Z.S/.

By hypothesis, all maps in End.H/ are Z.S/-linear and by Obseration 4.5 wehave D D QZ.S/. Any element s 2 S0 induces an endomorphism of H becauseS0 D

T1nD1.ZC p

nS/. Now let s 2 S such that �s 2 End.H/. Then it is easy tosee that pixis 2 piF C pixiZ. We infer that fais 2 fpiS C faiZ. Since ai isnot divisible by p, we have that s � zi modpiS for some integers zi . This showsthat s 2 S0.

Observation 4.7. If S D S0, then H is a free S -module of countable rank.

By Obseration 4.6, H is a module over S and H D fS ˚L1iD1 xiS .

Observation 4.8. Assume that

(a) EndZ.S/.S/ D EndZ.S/,

(b) Z.S/ ¤ S0 and QS is a division ring.

Then H is QE-transitive but not E-transitive.

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14 M. Dugas

If H is E-transitive, then by [10, Proposition 2.3]

D D Q.Z.End.H/// D QS0 D QZ.S/:

Since S0 and Z.S/ are pure in S , we get the contradiction S0 D Z.S/.

Observation 4.9. The subring S0 is (isomorphic to) a p-pure subring of Jp, thering of p-adic integers.

Observation 4.10. LetHn D fS ˚LniD0 fiS C xnC1Z. ThenHn is pure inH ,

Hn � HnC1 and H DS1nD1Hn.

Since S is p-reduced, the subring 1Z is p-pure in S . Let s 2 S0. Then thereexist elements zn 2 Z such that s � zn 2 pnS . It follows that znC1 � zn is anelement of pnS \ Z D pnZ for all n 2 N. Thus we have a p-adic Cauchy se-quence ¹znº which converges to a limit s� 2 Jp. It is easy to see that � W S0 ! Jpis an injective ring homomorphism with 1� D 1 2 Jp. It follows that .S0/� isa p-pure subring of Jp. We have:

Proposition 4.11. Let S be an E-ring of p-rank larger that 1 such that QS isa field. ThenH is QE-transitive but not E-transitive. Each finite rank S -submoduleis quasi-contained in a finite rank free S -module. If S is a strongly homogeneousE-ring of p-rank larger than 1, then H is QE-transitive, but not E-transitive.

Rings S with the properties mentioned in Proposition 4.11 exist in abundance.For example, let O be the ring of Gaussian integers, i.e. the ring of algebraic inte-gers in the quadratic field QŒi �. Let p D 13 and P D .3C 2i/O. Then S D OP ,the localization of O at the maximal ideal P , is a strongly homogeneous E-ring byresults due to Mader and Vinsonhaler [14].

5 Free modules over Dedekind domains

Remark 5.1. Let S be an integral domain and M an S -module. Then M is calledseparable, if for any finite subset F of M there exists a direct summand S of Msuch that F � S and S is a free S -module of finite rank. Let S be a subring of Qand M a separable S -module. Let X; Y be pure submodules of M of rank n. It iseasy to see that there is some automorphism ˛ of M such that ˛.X/ D Y .

In this context, the following is a classical result due to Erdos [7] (see also[8, Corollary 51.5]): Let F;F 0 be free abelian groups and H;H 0 pure subgroupsof F , resp. F 0. The following are equivalent:

(1) F=H Š F 0=H 0 and rank.H/ D rank.H 0/.

(2) There exists an isomorphism ˛ W F ! F 0 such that ˛.H/ D H 0.

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QE-transitive abelian groups 15

This shows that if F is a free abelian group of infinite rank, then there areplenty of pure, countable rank subgroups H;H 0 of F such that there is no auto-morphism of F mapping H onto H 0. We will show that this is quite differentfor endomorphisms of F . If V is some vector space and X; Y subspaces of Vwith dim.X/ � dim.Y /, then there exists a linear transformation � of V suchthat �.X/ D Y D �.V /. We will show that a similar result holds for free mod-ules over Dedekind domains.

The following is well known: LetM be a finitely generated, torsion-free moduleover a Dedekind domain D. Then there exists a natural number k � 0 such thatM Š Dk ˚ J , where J is a fractional ideal ofD and rank.M/ D k C 1 if J ¤ 0.Thus, if rank.M/ � 2, then M contains a free direct summand.

For abelian groups, the following lemma is due to Erdos [7] and can be foundin [8, Lemma 51.2].

Lemma 5.2. Let R be a Dedekind domain and F be a free R-module of infiniterank. Let H be a pure submodule of F of infinite rank. Then there exists a directsummand T of F such that T is a free submodule of H and rank.H/ D rank.T /.

Proof. Fix a basis B of F . Let S be a set of finite subsets of B .We say that S is a disjoint system for H if

(1) for all X; Y 2 S we have X D Y or X \ Y D ¿,

(2) for all X 2 S we have H \ spanR.X/ ¤ ¹0º.

Let � denote the set of all disjoint systems for H . Then � is partially orderedby set inclusion and it is obvious that � satisfies the hypothesis of Zorn’s lemma.Let S be a maximal element in �.

Assume that jS j < rank.H/ � jBj. Let

M DXX2S

spanR.X/ DMX2S

spanR.X/

since the elements of S are disjoint. Note that rank.M/ D jS j < rank.H/. LetC be the compliment of

SX2S X in B and N D spanR.C /. Then F DM ˚N .

Assume H \N D ¹0º. Then

H Š H=¹0º D H=.H \N/ Š .H CN/=N � F=N ŠM;

a contradiction to rank.M/ < rank.H/. Thus there exists some 0 ¤ g 2 H \Nand thus g 2 spanR.Y / for some finite subset Y of C . This shows S [ ¹Y º 2 �,a contradiction to the maximality of S in � and we have that jS j D rank.H/.

Now let

T DXX2S

.H \ spanR.X// DMX2S

.H \ spanR.X//:

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16 M. Dugas

Then rank.T / D jS j D rank.H/ and T � H . Moreover,

M=T DMX2S

spanR.X/=.H \ spanR.X//:

Since H is a pure submodule of F , each module spanR.X/=.H \ spanR.X/ isa finitely generated, torsion-free R-module for allX 2 S and thus projective sinceR is a Dedekind domain. This shows thatM=T is projective and thus T is a directsummand ofM and thus of F . Note thatH \ spanR.X/ is finitely generated sinceX 2 S is finite and R is a Noetherian ring.

Now we decompose the set S into two disjoint subsets S D S1 [ S2 such thatS2 D ¹X

0 W X 2 S1º and we write

T DMX2S1

Œ.H \ spanR.X//˚ .H \ spanR.X0//�;

where each module TX D .H \ spanR.X//˚ .H \ spanR.X0// is a torsion-free,

projective module of rank at least two. This shows that each TX contains a copyRXofR as a direct summand. Let T 0 D

LX2S1

RX . Then T 0 � H is a free summandof F with rank.T 0/ D rank.H/.

Theorem 5.3. Let R be a Dedekind domain, F be a free R-module and F1; F2 beinfinite rank submodules of F such that F1 is pure in F and rank.F1/ � rank.F2/.Then there exists ' 2 EndR.F / such that '.F1/ D F2 D '.F /.

Proof. Since F1 is pure in F , Lemma 5.2 yields a free direct summand T of Fsuch that T � F1 and rank.T / D rank.F1/ � rank.F2/. Let F D T ˚N and pickan � 2 Hom.T; F2/ such that � is surjective. Then � extends to a ' 2 End.F / bysetting '�N D 0.

6 @0-QE-transitive groups

In this section, let G denote a torsion-free @0-QE-transitive abelian group of in-finite rank, i.e. for any two pure, countable rank subgroups X and Y of G, thereexists some ' 2 End.G/ such that '.X/ � Y and Y='.X/ is torsion. Equivalently,the algebra Q End.G/ operates transitively on the set of all Q-subspaces of QGof countable dimension. As before, let D D ¹ı 2 EndQ.QG/ W ı D ı for all 2 End.G/º.

Proposition 6.1. LetG be a torsion-free @0-QE-transitive abelian group of infiniterank. If P is a pure, fully invariant subgroup of G of infinite rank, then P D G.

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QE-transitive abelian groups 17

Proof. The subgroup P contains a pure subgroup P 0 of countable rank. Let Ybe a pure, rank-1 subgroup of G. There exists a pure, countable rank subgroup Vof G with Y � V and there exists some ' 2 End.G/ such that '.P 0/ � V withV='.P 0/ torsion. Since '.P 0/ � P and Y \ '.P 0/ ¤ ¹0º, it follows that Y � P .This shows that P D G.

Corollary 6.2. Let G be a torsion-free @0-QE-transitive abelian group of infi-nite rank such that ¹0º is the only pure, finite rank fully invariant subgroup of G.Then G is irreducible and thus QE-transitive.

Proposition 6.3. LetG be a torsion-free @0-QE-transitive abelian group of infiniterank and let V be the purification of the sum of all fully invariant subgroups offinite rank of G.

(1) If rank.V / is finite, thenG0 D G=V is a torsion-free @0-QE-transitive abeliangroup of infinite rank and G0 is irreducible.

(2) If rank.V / is infinite, then G is countable and G is the union of an ascendingchain of finite rank, pure, fully invariant subgroups.

Proof. Since V is fully invariant inG, every ' 2 End.G/ induces a '0 2 End.G0/.Let X=V and Y=V be pure subgroups of G0 of countable rank. Then there existssome ' 2 End.G/ with '.X/ � Y and Y='.X/ torsion. Then '0.X=V / � Y=Vand .Y=V /='0.X=V / is torsion. This shows that G0 is @0-QE-transitive. Assumethat F=V is a finite rank fully invariant subgroup of G0. Then '0.F=V / � F=Vfor all ' 2 End.G/, i.e. '.F / � F for all ' 2 End.G/. This means that F is a fi-nite rank, fully invariant subgroup ofG and thus F � V . By Proposition 6.1,G0 isirreducible and thus QE-transitive. This shows that (1) holds. If V has infinite rank,then G contains a fully invariant, pure subgroup P of countable rank. By Proposi-tion 6.1, P D G has countable rank.

Theorem 6.4. Let G be a torsion-free @0-QE-transitive abelian group of infiniterank. If G is irreducible, then D D Q.

Proof. Assume that dimD.QG/ is infinite. Consider

X DM¹biZ˚ bidZ W i < !º and Y D

M¹biZ W i < !º;

where ¹bi W i < !º is a D-linearly independent subset of the group G. Note thatwe have Y \ Yd D ¹0º, where ¹bi W i < !º � G is linearly independent over Dand d 2 D �Q. Now there exists some ' 2 End.G/ such that '.X�/ � Y� witha torsion quotient. Then '.bid/ D '.bi /d 2 Y� \ Y�d D ¹0º and we have thecontradiction '.X/ D ¹0º. Thus we have D D Q or else dimD.QG/ is finite.

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18 M. Dugas

In the latter case, dimD.QG/ D 1, because in this case dimQ.D/ is infinite andthere are no surjectiveD-linear maps from a 1-dimensional space onto a 2-dimen-sional space.

Now assume that QG D D. We have that dimQ.D/ is infinite. Let 1 2 X; Ybe pure countable rank subgroups of G � D. Then there exists some ' 2 End.G/with '.X/ � Y and ' induces a D-linear map from D to D. Thus there existssome d 2 D with Xd � Y and d 2 Y . This means that X is contained in thesub-division ring S generated by Y . It follows that S D D and D is countableas well as G. Suppose there is some x 2 D that is transcendental over Q. It fol-lows that D is the rational number field Q.x/. Let X D QŒx2� and Y D QŒx4�.Obviously, X is not contained in the subfield QŒx4�. A contradiction. This showsthat all elements d 2 D are algebraic over Q. Let Y be a pure, countable sub-group of G. Then there exists some d 2 D \ End.G/ with Gd � Y . It followsthat D D Dd D QGd � QY � QG D D and thus Y D G for any pure, count-able rank subgroup of G, which is absurd.

Theorem 6.5. Let G be a torsion-free @0-QE-transitive abelian group of infiniterank. Then one of the following holds:

(a) G is irreducible and D D Q.

(b) There exists a pure, finite rank fully invariant subgroup V of G such thatG0 D G=V is irreducible and @0-QE-transitive.

(c) G is countable and is a union of an ascending chain of pure, finite rank fullyinvariant subgroups.

Proof. Let P be a pure, fully invariant subgroups of G of infinite rank. Let Lbe a pure subgroup of countable rank of P . Let Y be a pure subgroup of rank 1of G. Then there exists a pure, countable rank subgroup X of G such that Y � X .We infer that there exists some ' 2 End.G/ such that '.L/ � X and X='.L/torsion and thus Y=.Y \ '.L// is torsion. It follows that Y � '.L/ � P and weinfer that P D G andG is irreducible unlessG has pure, fully invariant subgroupsof finite rank. By Theorem 6.4, we have that D D Q if G is irreducible.

From now on, we may assume that whenever P ¤ G is a pure, fully invariantsubgroup of G, then rank.P / is finite. Let F be the set of all pure, fully invariantsubgroups of G. Then

PF is fully invariant and so is its purification V .

If V has finite rank, then (b) follows from Proposition 6.3 (1).If V has infinite rank, then (c) follows from Proposition 6.3 (2).

It is easy to see that case (b) actually occurs: LetG D Q˚ F , where F is a freeabelian group of infinite rank. ThenG is @0-QE-transitive but not irreducible whileG=Q is @0-QE-transitive and irreducible.

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QE-transitive abelian groups 19

Now we present an example for case (c): Let P D ¹pi W i < !º be a count-able list of distinct primes. Let Pn D ¹pi W n � i < !º and Sn D ZŒ 1

pW p 2 Pn�,

a subring of Q. LetG D

Mk<!

ekSk :

Let �k denote the type of Sk . Then G.�n/ DL0�i�n eiSi is fully invariant in G

and G DSn<! G.�n/. We need to show that G is @0-QE-transitive:

LetX be a pure, countable rank subgroup ofG. By [15, Theorem 1] there existsan infinite subset I of N such that

X D .X \ e0S0/˚

�Mi2I

hai i�

�with hai i� Š Si . This means that

ai DX0�j�i

ej sj i

for some sj i 2 Sj and si i ¤ 0. We call I the pivot set of X . There exist suitablenatural numbers ni such that

a0i DX0�j�i

ej sj ini DX0�j�i

ej s0j i

and all s0j i 2 Z. Let i > j . Then bi D a0i .�s0jj /C a

0is0j i 2 X has entry zero at

the ej . An easy induction shows that for all i 2 I there exist

bi D eiri CXi>j…I

ej rj i

such that X=X 0 is torsion, where X 0 DLi2I biSi . There exists a surjective func-

tion ˛ W I ! ! such that for all k < ! there is some k# 2 I with k D ˛.k#/ andk � k#. We define ' 2 End.G/ by

'.ei / D

´e˛.i/ for i 2 I;0 for i … I:

By our choice of the functions ˛, the homomorphism ' is surjective and for i 2 Iwe have '.bi / D '.eiri / D e˛.i/ri . This shows that G='.X/ is torsion. Now letY be a pure, countable rank subgroup of G. By Wang’s result [15], there existssome 2 Hom.G; Y / � End.G/ such that .G/ � Y with Y= .G/ torsion. Themap � D ı ' has the property that �.X/ � Y with Y=�.X/ torsion. This showsthat G is @0-QE-transitive. Of course,G is not QE-transitive since G is not homo-geneous.

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20 M. Dugas

We would like to thank the referee for suggesting the following

Remark 6.6. (1) For an uncountable cardinal n, nothing is known about n-QE-transitive groups.

(2) In [4, Definition 2.3] the notion of near isomorphism, a familiar term forfinite rank groups, was defined for torsion-free abelian groups of infinite rank. Thiscan be used to strengthen the notion of “n-QE-transitivity”, since near isomor-phism implies quasi-isomorphism. It might be worthwhile to take a look at this.

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Received June 25, 2013; revised November 15, 2013.

Author information

Manfred Dugas, Department of Mathematics, Baylor University,Waco, Texas 76798, USA.E-mail: Manfred_Dugas@baylor.edu