Torsion groups in cortorsion classes

Torsion Groups in Cotorsion Classes.

LUTZ STRU¨_NGMANN(*)

ABSTRACT - For torsion-free abelian groups Gof arbitrary rank we discuss the classRC(G) of all torsion groupsTsatisfying Ext (G, T)40 , that is the sub- class of all torsion groups of the cotorsion class cogenerated byG. The main question we consider is when for suchGthere exists a subgroup of the ratio- nals Z’R’Q such that RC(G)4RC(R).

Introduction.

Throughout this paper we work in the category Mod-Z of abelian groups. All terminology used here can be found in [F1], [F2] and [EM].

Cotorsion theories for abelian groups have been introduced by Salce in 1979 [S]. Following his notation we call a pair (F^,C) a cotorsion theory ifF^andCare classes of abelian groups which are maximal with respect to the property that Ext (F,C)40 for all FF^{, C}C^.

Salce [S] has shown that every cotorsion theory is cogenerated by a class of torsion and torsion-free groups where (F^,C) is said to be cogen- erated by the classA^ifC₄A^»_{4 ]}^XMod-Z_N^{Ext (A,X)}₄^{0 for allA} A( andF4^»(A^»)4 ]YMod -ZNExt (Y,X)40 for allXA^»(. Ex- amples for cotorsion theories are: (Mod -Z,D⁾4(^»(G^»),G^») withG4 4_p

5

_P^{Z(p) whereD}is the class of all divisible groups andPis the set of all prime numbers, (L^{, Mod -Z)}₄^{(»(Z»),Z») where}Lis the class of all free groups, and the classical one (RF^,C O⁾₄^(»(Q»),Q^») whereRF^is

(*) Indirizzo dell’A.: The Hebrew University, Department of Math. Givat Ram, Jerusalem 91904, Israel. E-mail: lutzHmath.huji.ac.il

The author was supported by a MINERVA fellowship.

the class of all torsion-free groups andC Ois the class of all (classical) co- torsion groups. In view of the last example the classesF^andCof a cotor- sion theory (F,C) are said to be the torsion-free class and the cotorsion class of this cotorsion theory.

In this paper we shall restrict to cotorsion classes cogenerated by a single torsion-free groupG and turn our attention to the subclass of all torsion groups of the cotorsion class cogenerated by G. This class was denoted by RC(G) and first studied by Wallutis and the author in [SW].

A characterization of these classes is obviously closely related to the solution of the Baer problem (e.g. see [F2]); put into our context it says thatRC(G) is maximal, i.e.RC^(G)₄^{F, whereF}is the class of all torsion groups, if and only if G is free.

Recall that torsion-free groups of rank-1 can be indentified with sub- groups of the rationalsQ and are therefore also called rational groups.

Using a result by Salce [S] a full characterization of RC(R) for some given rank-1 group was obtained in [SW]. Introducing the quasi-reduced typetype^qr(R) of an rank-1 groupRit was shown thatRC^(R)₄^RC^(R₈⁾ if and only iftype^qr(R)4type^qr(R8) for any rank-1 groupsR,R8; when representing the type ofR bytype(R)4(r_p)_pP the quasi-reduced type of R can be represented by type^qr(R)4(s_p)_pP with s_p4r_p whenever r_p40 orr_p4 Qands_p41 otherwise. This characterization of the classes RC(R) for rank-1 groups had immediate consequences for completely decomposable groups. Therefore, it is of particular interest to know when the classRC(G) for a given torsion-free groupGis generated by a rational group R, i.e. RC^(G)4RC(R) for some rational group R.

After we shall have considered some basic facts in section 1 we define the classRof all torsion-free groupsGsuch thatRC(G) is generated by a rational group, i.e. R_{4 ]}^G torsion-free :RC^(G)₄^RC(R) for some ratio- nal group R’Q(. We will first show in section 2 that all torsion-free groups of finite rank belong toR. In fact we are able to prove that any tor- sion-free group G of finite rank satisfies RC^(G)₄^RC( OT (G) ) where OT (G) denotes the outer-type ofG. Recall that the outer-type ofGis de- fined to be the supremum of all rank-1 torsion-free quotients ofG. We shall use this result to show that for almost all rational groupsRthere is an in- decomposable almost completely decomposable group G of arbitrarily large rank satisfyingRC^(G)₄^RC(R). Moreover, we show that ifR is di- visible by at least one primepthen there is even an indecomposable homo- geneous group Gof arbitrarily large rank with RC^(G)₄^RC(R). Finally,

for any rational groupR which is non-idempotent (i.e. End (R)`OR) an indecomposable group G of given rank n is constructed such that G is homogeneous of typeR and RC(G)4RC(R). This shows that Griffith’s solution of Baer’s problem (see [G]) cannot be generalized to homoge- neous groups of non-idempotent type.

In section 3 we consider torsion-free groupsGof countable rank and extend the notion of outer-type in a suitable way; OT_G is defined to be the set of all quasi-reduced outer types of finite rank pure subgroups of G. Using this notion a criterion is given when GR. A countable tor- sion-free group G satisfies RC^(G)₄^RC(R) for a rational group if and only if the supremum of OT_G exists and OT_G is Q-closed which means that wheneverPis an infinite set of primes and for allpP there exists ROT_G such that xRp( 1 )4 Q, then there exists ROT_G such that xpR( 1 )c0 for almost all pP. It turns out that the classRis no longer closed under direct summands if we restrict to countable groups.

Finally, we prove in section 4 a very technical necessary and suffi- cient condition for torsion-free groups of arbitrary rank to belong to R^. Therefore, we extend the notion of Q-closed to a condition on the van- ishing of certain Ext-groups.

Let us begin with some easy facts.

1. Preliminaries.

In this section we first recall the definition of the class RC^{(G) for a} given groupGas it was given in [SW]. Moreover, we introduce two relat- ed classes of groups and recall some easy and known results adjusted to these notations.

For an arbitrary group G the classes G^» and ^»(G^») are defined by G^»4]XMod-ZNExt(G,X)40( and ^»(G^»)4 ]YMod-ZNExt(Y, X)4 40 (XG^»(, respectively. Due to Salce [S] the pair (^»(G^»),G^») is called thecotorsion theory cogenerated by G. In view of the classical co- torsion theory (RF^,C O^{) where} C O₄^Q» is the class of all cotorsion groups and RF₄^»(Q») is the class of all torsion-free groups, we also callG^»thecotorsion class cogenerated by G. As we have seen in the in- troduction it makes sense to restrict our attention to the cotorsion class- es cogenerated by torsion-free groups and furthermore to its subclass of all torsion groups.

LetFbe the class of all torsion groups. In [SW] the classRC^{(G) was} defined for any group G as RC^(G)₄^G»OF, the class of all torsion

groups belonging to the cotorsion classG^»cogenerated by G. It is well known thatG^»as well asFare closed under epimorphic images and ex- tensions (see [F1] and [S]). Hence we immediately have the following (see also [SW]):

LEMMA 1.1. For any group G the following are true:

(i) RC^(G) is closed under epimorphic images;

(ii) RC^(G) is closed under extensions, especially under finite di- rect sums;

(iii) if G is torsion-free, thenRC^(G) contains all torsion cotorsion groups, i.e. RC^(Q)4C OOF’RC^(G).

Moreover, an easy and well-known lemma is sometimes very useful.

First recall that the basic subgroup Bof a torsion groupTis the direct sum B4_p

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_P^Bp of the basic subgroups B_p of the p-components T_p; for each primep,Bpis a direct sum of cyclicp-groups,Bpis a pure subgroup ofT_p and the quotient T_p/B_p is divisible (see [F1]). The proof of the fol- lowing result is left to the reader (see also [SW]).

LEMMA 1.2. Let T be a torsion group and B’T a basic subgroup of T. Then, for any group G,T is an element ofRC^(G)if and only if B is.

In [SW] the class RC(G) was studied for torsion-free abelian groups G, in particular for subgroups R of the rationals Q and for completely decomposable groups. Recall that thequasi-reduced type type^qr(R) of a type R is defined as the subgroup S of Q such that xSp( 1 )41 if 0c cxRp( 1 )cQ and xpS( 1 )4xpR( 1 ) else (p a prime).

LEMMA 1.3 (Strüngmann-Wallutis, [SW]). Let R be a rational group withx(R)4(r_p)_pPand let T4_pP

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^Tpbe a reduced torsion group with p-components T_p.

Then Ext (R,T)40 if and only if the following conditions are satisfied:

(i) Tp is bounded for all p such that rp4 Q; (ii) Tp40 for almost all p such that rpc0 .

In particular, RC^(R)₄^RC^{( typeqr(R) )} and for two rational groups R and S we have RC^(R)’RC^(S) if and only if type^qr(R)F Ftype^qr(S).

It was shown in [SW] that any countable torsion-free abelian groupG satisfies RC^(G)₄^RC(C) for some completely decomposable group C. Moreover, for a large class of torsion-free groupsG it was proved that RC^(G)4RC(R) for some rational group R. Motivated by these results we introduce the following classes of torsion-free abelian groups.

DEFINITION 1.4. Let RF be the class of all torsion-free abelian groups. Then

(i) ÷4 ]GRF^:RC^(G)4RC^(C) for some completely decompos- able group C(;

(ii) R_4]GRF^:RC^(G)₄^RC^(R) for some rational group R’Q(; (iii) RS_{4 ]}^GRF^{: R}C^(G)₄^RC^{(Tst(G) )}₍^, where Tst(G) denotes the typeset of G and RC^{(Tst(G) )}₄^RC

g

_RTst(G)

⁵

^R

h

^.

Clearly R^’÷and RS^’÷and it was shown in [SW] that any count- able torsion-free group belongs to ÷ but it is undecidable in ZFC whether or not ÷4RF. Moreover, the following is a collection of the main results in [SW]. Recall that a torsion-free groupGis a Butler group if Bext¹(G,T)40 for all torsion groups T (for details on the functor Bext¹ and Butler groups see [F3]). Moreover,G satisfies the torsion ex- tension property if for every pure subgroup H of G and every torsion group T the induced homomorphism Hom (G,T)KHom (H,T) is surjective.

LEMMA 1.5 (Strüngmann-Wallutis, [SW]). The following hold.

(i) the class RS contains all Butler groups of arbitrary rank;

(ii) the class Rcontains all torsion-free groups of finite rank sat- isfying the torsion extension property, hence all Butler groups of finite rank.

Nevertheless, the next example which can be found in [F2, Example 5 on p. 125] shows that the classRSis ugly although it contains the nice subclass of all Butler groups.

EXAMPLE 1.6. Let pbe a prime and pa p-adic integer, not a ratio- nal; say p4s₀1s₁p1R1s_npⁿ1R with 0Es_iEp. Moreover, let x₁,x₂ be linearly independent elements and, for each nN, let y_n4p²ⁿ(x₁1(s₀1s₁p1R1s_n21pⁿ²¹)x₂).

We define G by

G4ax₁,x₂,y₁,y₂,R,y_n,Rb’Q^(p)x₁5Q^(p)x₂.

It can be easily verified that G is indecomposable and homogeneous of typeZ(see [F2]). Also,Gis not a Butler group since homogeneous But- ler groups are completely decomposable.

Moreover, we haveRC^(G)₄^RC^(Q(p)) (see [SW, Example 4.5]), hence G÷butGRS^{, elseG}would be free by Griffith’s solution of the Baer problem (see [G]).

Now put H4G5Q^(p), then Tst (H)4 ]Z,Q^(p)( and clearly RC^(H)₄^RC^(Q(p))₄^RC( Tst (H) ), hence HRS^.

COROLLARY 1.7. The class RS is neither closed under direct sum- mands nor extensions.

PROOF. In the Example 1.6 the groupHRSbut its direct summand Gdoes not belong toRS. Moreover, the groupGfrom Example 1.6 shows that RS is not closed under extensions. r

Since the group Gfrom Example 1.6 is not a Butler group it follows that HRS is not a Butler group and hence the class RS strictly con- tains the class of all Butler groups which answers Question 4.7 from [SW] negatively.

We are now ready for the next section in which we will consider tor- sion-free groups of finite rank and show that the classR behaves much nicer than RS^.

2. Torsion-free groups of finite rank.

In this section we shall first show that the classR contains all finite rank torsion-free groups. Let us begin with an easy lemma.

LEMMA 2.1. The following are equivalent, where RF F denotes the class of all torsion-free groups of finite rank:

(i) RORF F is the class of all torsion-free groups of finite rank;

(i) RORF F is closed under extensions;

(iii) RORF F is closed under subgroups.

PROOF. The implications (i)¨(ii) and (i)¨(iii) are trivial. To show (iii)¨(i) notice that any finite rank torsion-free group can be embedded into its divisible hull which is obviously a member ofR. The remaining implication (ii)¨(i) is easily seen by induction on the rank and left to the reader. r

To continue we need Baer’s Theorem from [B].

LEMMA 2.2 (Baer, [B]). Let T be a torsion group and G a torsion- free group such that Ext (G,T)40 . Then the following hold:

(i) if ]p1,R,pi,R( is an infinite set of different primes for which p_iTET,then G contains no pure subgroup S of finite rank such that G/S has elements c0 divisible by all p_i;

(ii) if, for some prime p,the reduced part of the p-component of T is unbounded, then G contains no pure subgroup S of finite rank such that G/S has elements c0 divisible by all powers of p.

Moreover, if G is countable, then (i)and(ii)suffice for Ext (G,T)to be zero. r

We obtain the following lemma.

LEMMA 2.3. Let G be a countable torsion-free group and H a pure subgroup of G of finite rank. If T is a torsion group such that TRC^(G), then TRC^(G/H).

PROOF. LetH’GandTbe given as stated such thatTRC^{(G). As-} sume that TRC(G/H), then since G/H is torsion-free there exists a pure subgroupK/HofG/Hof finite rank such that (G/H) /(K/H) contains an element x violating one of the conditions from Lemma 2.2. But (G/H) /(K/H)`G/K is torsion-free, henceGcontains a pure subgroupK of finite rank such thatG/K contains an element violating one of Baer’s conditions. Since Gis countable we conclude that Ext (G,T)c0 , hence TRC(G) - a contradiction. r

THEOREM 2.4. Let G be a torsion-free group of finite rank. Then RC^(G)₄^RC^(R) for some rational group R.

PROOF. We induct on the rank n of G. If G has rank one, then the claim is trivially true. Hence assumeGhas ranknD1 . We choose a pure

subgroup H of G of rank n21 and obtain G/H4S’Q. Now clearly RC^(S)ORC^(H)’RC^{(G) and R}C^(G)’RC^{(H) .}

But by Lemma 2.3 we also have that RC^(G)’RC(S), hence we obtain RC(G)4RC(H)ORC(S). By induction hypothesisRC(H)4RC(R8) for some rational group R8. Thus RC^(G)4RC^(S)ORC^(R8)4RC^(R) where R4SOR8’Q. r

The next theorem shows that we can chooseR in Theorem 2.4 to be the outer type ofG. Recall that for a torsion-free group Gof finite rank theouter typeOT (G) is defined as follows: Letg₁,R,g_n be a maximal linearly independet subset ofGand putH_i4ag₁,R,g×_i,g_i11,R,g_nb*’

’G, where g×means that the element is missing. Then put S_i4G/H_i’Q and define OT (G)4sup]S1,R,Sn(. The set ]Si: iGn( is called the cotypesetofG. Note that for a pure subgroupHofGwe have OT (G)F FOT (H). Similarly, the inner type IT (G) of G is defined as IT (G)4 4inf]ag_ib*:iGn(.

THEOREM 2.5. Let G be a torsion-free group of finite rank and OT (G) its outer type. Then RC^(G)₄^RC( OT (G) ).

PROOF. By Theorem 2.4 we know thatRC^(G)₄^RC(R) for some ra- tional groupR. We induct on the ranknofGto show that we can choose R4OT (G). For rational groups S it is trivial since OT (S)4S. Hence assume G is of rank nD1 . We let Si be defined as in the definition of OT (G). Since G is of finite rank we obtain by Lemma 2.3 RC^(G)4 4RC^(R)’RC^(Si) for every iGn and therefore RC^(G)₄^RC^(R)’

’_i

1

_Gn^RC(Si)4RC(OT (G) ) since OT (G) is the supremum of the typesS_i. Now assume thatRC(R) is strictly contained inRC(OT (G) ), then there is TRC(OT (G) )0RC(R). Thus TRC(Si) and hence TRC(Hi) for any iGn. But OT (G)FOT (H_i), hence RC^{(OT (G) )’R}C^{(OT (H}i) )4 4RC^(Hi) by induction hypothesis and thereforeTRC^(Hi) - a contradic- tion. r

Using a result due to Warfield (see [W]) we can determine the outer type and hence RC(G) explicitly for a torsion-free group G of finite rank.

LEMMA 2.6 (Warfield, [W]). Let G be a torsion-free group of finite rank and let F be a free subgroup of G such that G/F is torsion, i.e.

G/F4

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_p ^{Tp with Tp}4Z(p^{ip, 1})5R5Z(p^ip,np)

and 0Gi_{p, 1}GRGi_p,npG Q. Then OT (G)4[ (i_p,np) ] and IT (G)4 4[ (i_p,n1) ].

We will now show that for almost all typesR the classRC^{(R) can be} realized asRC(G) for an indecomposable, almost decomposable group of rank nfor any natural number n. This proves that the structure of the groupGis less effected byRC(G) than for example byG^»even for finite rank groups in RSOR^.

LEMMA 2.7. If Q`OR is a type such that R`OQ^(P) for any set P of primes of cardinality at most two, then there exists for any natural number n an indecomposable, almost decomposable group G of rank n such that RC^(G)₄^RC^(R).

PROOF. Forn41 there is nothing to prove taking G4R. Hence as- sume thatnD1 . We may assume without loss of generality that Ris in reduced form, i.e. R4type^qr(R). If R`OQ<sup>(P)</sup> for any finite set P of primes, then we can give an explicite example. We will use the construc- tion of an almost completely decomposable group as given in [A, Example 2.2]. Hence it is enough to find incomparable types A<sub>i</sub> (iGn) such that the supremum of theA<sub>i</sub>’s is exactlyR. ButR`OQ^(P)for any fi- nite setP of primes implies that there are infinitely many primespsuch thatxpR( 1 )4 Qor there are infinitely many primespsuch thatxpR( 1 )4 41 . LetPbe the set of these primes. We dividePinndisjoint infinite sub- sets P_i and define the types A_i by letting

xpA_i( 1 )4xpR( 1 ) for pP_i x_p^Ai( 1 )40 for p_j

0

_ci^Pj

xpA_i( 1 )4xpR( 1 ) else .

Since Q`OR we may assume without loss of generality that there is a primepsuch thatx_p^Ai( 1 )40 for alli. Then clearly the supremum of the A_i’s isRand by [A, Example 2.2] there exists an indecomposable almost completely decomposable groupGof ranknsuch that Tst (G) is the meet

closure of ]A₁,R,A_n(, hence Lemma 1.5 implies RC^(G)₄^RC^{(Tst (G) )}₄^RC^{(R) .}

If R`Q^(P) for some finite set P of primes of cardinality greater than two, then the existence of the desired almost completely decomposable group follows from [AD, Theorem 1.8]. DividePinto two non-empty sets P1 and P2 and choose pP2. Put T to be the meet closure of the set ]Q^(P1),Q^(P2),Q^(p)(, then [AD, Theorem 1.8] gives the existence of an al- most completely decomposable groupGof any finite ranknsuch that the critical typeset of G is contained in ]Q^(P1),Q^(P2),Q^(p)( and hence RC^(G)₄^RC^(R). _r

Note that if R`OQ^(P) for some set of primes of cardinality less or equal to two, then it is known that an indecomposable almost completely decomposable groupGwithRC^(G)4RC(R) must have rank at most two by the structure of the critical typeset (see also Lemma 2.11 and [A, The- orem 2.3]).

If we don’t requireGto satisfyRC^(G)₄^RC(Tst (G) ), then we can re- alize any typeQ`OR which is divisible by at least one prime (in order to exclude Z) as the outer type of an indecomposable homogeneous group.

LEMMA 2.8. LetQ`OR be a type such that pR4R for at least one prime, then there exists for any natural number n an indecomposable homogeneous torsion-free group G such that RC^(G)₄^RC^{(R). More-} over, if S is the type of G, then S7Q^(p)4R.

PROOF. For n41 the claim is trivial, hence assume nD1 . Fix the prime p for which pR4R and let S be the type identical with R but xpS( 1 )40 . We now use a known example (see [F2, Example 5 on p. 125]).

Let H4_i

5

_Gn^Q(p)ai be completely decomposable and homogeneous of typeQ^(p) and of rank n. Chosse n21 algebraically independent p-adic units p2,R,pn and let p141 . For m41 , 2 ,R we put

x_m4p^2m(a₁1p2ma₂1R1pnma_n)H

wherepim4s_i01s_i1p1R1s_{i(k21 )}p^kis the (k21 )st partial sum of the standard form of pi. We define A as

A4aa1,R,an,x1,R,xk,Rb.

Then A is indecomposable and homogeneous of type Z. Moreover, for each subgroupBof rankn21 we haveA/B`Q<sup>(p)</sup>. Furthermore,Astays indecomposable if it is tensored by any rank one groupLsuch thatpLc cL. We putG4S7Ato obtain a torsion-free group of ranknwhich is ho- mogeneous of type S and for each subgroup C of rank n21 we have G/C`S7Q^(p)`R, hence OT (G)4R and therefore RC^(G)₄ 4RC^(R). _r

ForQwe have a similar realization lemma. Recall that a torsion-free groupG of finite rank is calledalmost-freeif any proper pure subgroup of G is free.

LEMMA 2.9. For each natural number n there is an indecompos- able almost-free group G of rank n such that RC^(G)₄^RC^(Q).

PROOF. By a construction due to Corner (see [F2, Exercise 8 on p.

128]) there exists an indecomposable torsion-free groupGof ranknsuch that G is almost-free and each quotient of rank 1 is divisible, hence OT (G)4Q and we are done. r

Note that by the construction in Lemma 2.8 the type ofGis as close toR as possible sinceG homogeneous of type R would imply IT (G)4 4OT (G) and hence [A, Corollary 1.13] implies thatGis completely decom- posable. But if we assume that the typeRis non-idempotent, then we can even do better. Recall that a type R is idempotent if End (R)`R.

LEMMA 2.10. Let R be any non-idempotent type. Then there exists for any natural number n an indecomposable group G of rank n which is homogeneous of type R and RC^(G)₄^RC^(R).

PROOF. Since R is non-idempotent there exists an infinite set P of primes such that 0Ex^R_p( 1 )E Q. PutSto be the type such thatx^S_p( 1 )4 4x^R_p( 1 ) wheneverpPandx_p^S( 1 )4x_p^R( 1 )11 ifpP. We will show that there is a torsion-free groupGof ranknsuch thatGis homogeneous of typeRand OT (G)4S, henceGis as desired. By an easy argument (see e.g. Schultz [Sch]) it is enough to prove that there exists a torsion-free groupHof ranknsuch thatHis homogeneous of type Zand the outer type ofHisQwhereQ4a1 /p: pPb. Without loss of generality we may assume thatP4P. LetF4_i

5

_Gn^Zbe a free group of rankn. We choose n-tuples t_i4(a₁ⁱ,a₂ⁱ,R,a_nⁱ)F such that

(i) gcd (a₁ⁱ,a₂ⁱ,R,a_nⁱ)41 ; (ii) Naji

NGnfor alljGn, whereNaNdenotes the absolute value ofa. LetP4 ]p_i:iv(be an enumeration of the set of primes such that p_iis thei’th prime number. PutG4F1at_i/p_i: p_iPb. Then Gis a tor- sion-free group of ranknand it remains to show thatGis homogeneous of typeZ, indecomposable and has outer typeQ. The later is easily seen and hence left to the reader. To prove thatGis homogeneous of type Z let g4(g₁,g₂,R,g_n)G (viewed inside its divisible hull) and assume thatgis divisible by infinitely many primes. Without loss of generalityg is divisible by allpiP. Thengftimodulo piGfor all iv. Thus with- out loss of generalityg12a1ⁱc0 is divisible by pi. But this can happen only for finitely manyisincea₁ⁱis less or equal tonbutp_iis the order of n˜log(n), where log denotes the logarithm with basis 10 - a contradic- tion. Therefore,Gis homogeneous of typeZ. It remains to prove that G is indecomposable. Assume that G4G₁5G₂ with G_ic0 and note that G/F`_i

5

_v^{Z(pi). Thus F}4F15F2 with Fi4FOGi and hence G/F4 4G1/F15G2/F2. But then almost all elementst_imust belong either toG1

or G₂ which is obviously a contradiction (¹). r

As a corollary we obtain a general version of Griffith’s solution of the Baer problem for groups of finite rank.

COROLLARY 2.11. Let G be a torsion-free group of finite rank and homogeneous of the idempotent type R.Then G is completely decompos- able if and only if RC^(G)₄^RC^(R).

PROOF. One implication is trivial, hence assume that RC^(G)4 4RC^{(R). Since R}C^(G)4RC(OT (G) ) we obtain that the quasi-reduced types ofRand OT (G) are equal by Lemma 1.3 but sinceRis idempotent this implies that the typesR and OT (G) are equal, hence IT (G)4R4 4OT (G) and the result follows by [A, Corollary 1.13]. r

COROLLARY 2.12. Let G be a torsion-free group of finite rank. Then G is free if and only if RC^(G)₄^RC^(Z).

PROOF. Again one implication is trivial, hence assume thatRC^(G)₄

(¹) The author would like to thank Prof. C. Vinsonhaler for indicating the con- struction to him.

4RC(Z). To apply Corollary 2.11 we have to show thatGis homogeneous of typeZ. But RC^(G)₄^RC(Z) implies that OT (G)4Z, hence Gmust be homogeneous of type Z. r

Let us note that Corollary 2.11 may fail if we don’t assume that Ris idempotent (see Lemma 2.10), hence Griffith’s solution of the Baer prob- lem cannot be generalized to homogeneous groups of non-idempotent type.

Our final well-known example shows that even for the class of finite rank torsion-free abelian groups we cannot obtain some kind of Krull- Schmidt theorem using RC^.

EXAMPLE 2.13 (Jonsson, [J]). There are indecomposable torsion- free groups A₁,A₂and B₁,B₂of ranks 1 , 3 , 2 , 2 ,respectively such that A₁5A₂4B₁5B₂ but RC^(Ai)cRC^(Bj) for i,j]1 , 2(.

PROOF. The example is well-known and it is easy to check that the constructed groupsA_i and B_i fori41 , 2 are almost completely decom- posable, hence members of the class RS. Since the four groups contain types of the formQ^(p)for different primes p it follows immediately that RC^(Ai)cRC^(Bj) for i,j]1 , 2(. r

3. Torsion-free groups of countable rank.

In this section we shall consider torsion-free groups which are count- able and we will first characterize those which belong to the class R^. LEMMA 3.1. Let G be a countable torsion-free group and T a tor- sion group. Then TRC^(G)if and only if TRC^(H)for any finite rank pure subgroup H of G.

PROOF. Clearly TRC(G) implies that TRC(H) for any finite rank pure subgroup H of G. Conversely, assume that TRC^(H) for every pure subgroup H of G of finite rank. If TRC^{(G), then} there exists a pure subgroup H of finite rank of G and an element x1HG/H violating one of the two conditions of Baer’s Lemma 2.2. Note that here we need the countability ofG. We write ax1Hb*’

’G/H as

ax1Hb*4H8/H

for some pure subgroup H8 of G of finite rank. Then Lemma 2.2 implies that Ext (H8,T)c0 , hence TRC^(H₈) - a contradic- tion. r

For our next result we need the following definition which is a rea- sonable extension of the notion of outer type to groups of countable rank.

DEFINITION 3.2. Let G be a countable torsion-free group. Then we define the outer type class OT_G of G as

OTG4 ]type^qr(OT (H) ) : H a pure subgroup of G of finite rank(. In the following theorem RC^(OTG) means RC

g

_R

⁵

_OT

G

R

h

^.

THEOREM 3.3. Let G be a countable torsion-free group. Then RC^(G)4RC^(OTG). Hence GR if and only if _R

5

_OT

G

RR^.

PROOF. By Lemma 3.1 we have that RC^(G)₄ORC^{(H) where H} ranges over all pure subgroups ofGof finite rank. But by Theorem 2.5 it follows that RC^(H)4RC^{(OT (H) )}4RC^{( typeqr}(OT (H) ) ) for every pure subgroup H of G of finite rank. Hence

RC^(G)₄ORC^{(OT (H) )}₄ORC^{( typeqr}(OT (H) ) )4RC^(OTG) which finishes the proof. r

An easy corollary is the well-known result of Pontryagin.

COROLLARY 3.4. Let G be a countable torsion-free group. Then G is free if and only if each pure finite rank subgroup of G is free.

PROOF. One implication is trival, hence assume that any pure sub- groupHofGof finite rank is free. Then OT (H)4Z, hence by Theorem 3.3RC^(G)₄^RC^(OTG)4RC(Z) and thus Gis free by Griffith’s solution of the Baer problem (see [G]). r

Theorem 3.3 reduces the question of whether or not a countable tor- sion-free groupGbelongs toRto completely decomposable groups with types in quasi-reduced form. Note that the supremum of infinitely many types need not exist as the following example demonstrates.

EXAMPLE 3.5. LetPi(iv)be a system of disjoint infinite subsets of the primes P. Put R_i4a1 /p:pPib’Q. Then the types R_i (iv) have no supremum.

We need another definition.

DEFINITION 3.6. Let S4 ]R_i:iI( be a set of types. Then S is called infinity-closed(Q-closed)if for any infinite set P of primes such that for all pP there exists iI such thatxpRi( 1 )4 Q,there exists iI such that x^R_pⁱ( 1 )c0 for almost all pP.

PROPOSITION 3.7. Let C be completely decomposable with types in quasi-reduced form. Then CR if and only if S4sup (Tst (C) ) exists and Tst (C) is Q-closed. In this case RC^(C)₄^RC^(S).

PROOF. If the supremumS of Tst (C) exists and is Q-closed, then it is easily checked thatRC^(S)₄^RC(C) using that all the types of C are quasi-reduced. Conversely, if RC^(C)₄^RC(R) for some rational group R, then we first have to show that the supremum of Tst (C) exists. With- out loss of generality we may assume that R is in quasi-reduced form.

Hence Lemma 1.3 implies that RFS for any type STst (C) since all types are quasi-reduced. Now assume that R is not the supremum of Tst (C). Then there exists a type L such that LER and LFS for all types STst (C). Therefore there exists

(i) an infinite set of primes P such that xpR( 1 )41 and xpL( 1 )40 for all primes pP or

(ii) there exists a prime p such that xpR( 1 )4 Q and xpL(1)EQ. In both cases we easily obtain a contradiction since RC^(C)₄ 4RC(Tst (C) ) and LFS for every type STst (C). Thus sup (Tst (C) ) exists and clearly Tst (C) must be Q-closed. r

COROLLARY 3.8. Let G be a countable torsion-free group. Then G R if and only if sup (OTG) exists and OTG is Q-closed. r

PROOF. The claim follows by Theorem 3.3 and Proposition 3.7. r

It is now trivial to see that for groupsG÷we also haveGR^{if and} only if sup (Tst (G) ) exists and Tst (G) is Q-closed.

Finally let us remark that in contrast to the finite rank case for

countable groups also the behavior of the classRis as ugly as the behav- ior of the class RS as the following examples show.

EXAMPLE 3.9. Let Pi (iv) be disjoint infinite sets of primes. Put Ri4a1 /p: pPib’Q and let R4a1 /p:pPi(iv)b’Q. Then the supremum of the typesR_i (iv) does not exist but if we adjoin R, then the supremum of ]R,R_i: iv( exists and is equal to R.

EXAMPLE 3.10. Enumerate all primes by v, e.g. P4 ]p_i: iv( and let Ri4a1 /(p_iⁿ) :nvb’Q for iv. Moreover, let R4a1 /p:p Pb’Q,then the set ]Ri:iv(is notQ-closed but the set]R,Ri:i v( is Q-closed.

Thus a set of types which is Q-closed may contain a subset which is notQ-closed and the same holds for sets of types such that there supre- mum exists. We obtain the following corollary which contrasts the finite rank case.

COROLLARY 3.11. The class of all countable torsion-free groups G R is neither closed under direct summands nor epimorphic images nor extensions.

PROOF. We take the typesRi(iv) andRfrom Example 3.9 or from Example 3.10 and putC4_i

5

_v^{Ri. ThenCR}by Proposition 3.7. ButC5 5RR, hence the claim follows. r

4. Torsion-free groups of arbitrary rank.

In this section we consider torsion-free groups of arbitrary rank and try to generalize the results from Section 2 and Section 3, i.e. we are looking for a characterization of groupsGR. We don’t have a charac- terization in terms of types any longer as we had in the finite rank and coutable case but we can characterize the groups by some conditions on certain extension groups. Let us first recall a basic theorem from [SW]

which characterizes the classes of torsion groups that can appear as RC(C) for some completely decomposable group C.

THEOREM 4.1 (Strüngmann-Wallutis, [SW]]. Let÷be a class of tor- sion groups. Then ÷4RC^(C) for some completely decomposable group C if and only if the following conditions are satisfied:

(i) ÷ contains all torsion cotorsion groups;

(ii) ÷ is closed under epimorphic images;

(iii) _nv

5

^Z(pn)÷ if and only if ÷ contains all p-groups for all primes p;

(iv) If P is an infinite set of primes then,_p

5

_P^Z(p)÷if and only if _p

5

_P^Tp÷ for all p-groups Tp÷;

(v) If P is an infinite set of primes such that _p

5

_P^{Z(p)÷ then}

there exists an infinite subset P8of P such that_p

5

_X^Z(p)÷for all infi- nite X’P8.

PROOF. See Theorem 3.6 in [SW]. r

We will now first give a homological characterization of the groups in RS which clarifies the connection to Butler groups. We therefore define the class B(G) for a torsion-free group G as B^(G)_{4 ]}^{T torsion} : Bext¹(G,T)40(.

THEOREM 4.2. Let G be a torsion-free group. Then GRS ^{if and} only if RC^{(Tst (G) )’}B^(G).

PROOF. Let G be a torsion-free group such that GRS^{. Let C}4 4_gG

5

^agb_*. We obtain the following exact sequence

0KKKCKGK0

where the mappingCKGis the obvious one andKis the corresponding kernel. In fact, this sequence is easily checked to be balanced exact and hence we obtain the induced sequences

Hom (C,T)KHom (K,T)KBext¹(G,T)KBext¹(C,T)40 and

Hom (C,T)KHom (K,T)KExt (G,T)KExt (C,T)

with the same mapping Hom (C,T)KHom (K,T) for any torsion group T. IfTRC(Tst (G) ), then Ext (C,T)40 , hence also Ext (G,T)40 and thus the mapping Hom (C,T)KHom (K,T) is surjective, i.e.

Bext¹(G,T)40 . Hence RC^(C)’B(G) as claimed. Conversely, assume thatRC^(C)’B(G). In order to prove thatGRSwe have to show that

RC^(C)’RC(G) to proveGRS. Therefore letTRC(C). By assumption we obtain Bext¹(G,T)40 , hence again the mapping Hom (C,T)K KHom (K,T) is surjective and it follows that Ext (G,T)’Ext (C,T) which is zero by the choice of T. Hence Ext (G,T)40 and therefore TRC^(G). _r

An immediate corollary is the following. Recall that a torsion-free group G is a B1-group if B^(G)4F.

COROLLARY 4.3 (Strüngmann-Wallutis, [SW]). Let G be a B₁-group.

Then GRS^.

QUESTION 4.4. To characterize the Butler groups among the groups in RS one need to know the following: which is the minimal class of torsion groups 3 such that 3’B^(G) forces G to be a Butler group?

Our next theorem will give a characterization of the classes that can appear as RC(R) for some rank-1 group R by adding two more condi- tions in Theorem 4.1. Therefore we need the following definition.

DEFINITION 4.5. Let÷ be a class of torsion groups and P1 and P2

two subsets of the primes.

(i) P_{4 ]}^P’P (P infinite)N_p

5

_X^Z(p)÷ for all infinite X’P(; (ii) We say that P1 and P2 are equivalent(P₁AP2) if their sym- metric difference P1DP2 is finite;

(iii) For PP ^{we let}

[P]4 ]QP^:P_A^Q₍ be the equivalence class of P inside P^;

(iv) M÷4 ][P] :PP₍;

(v) For [P], [Q]M÷ we define [P]’[Q] if and only if P is al- most contained in Q,i.e. P0Q is finite; hence (M÷,’)is a partially or- dered set.

Note that the definition of ’ in (v) is well-defined and so is a finite union of equivalence classes _i

0

Gn[P_i]4

k

_i

0

GnP_i

l

. But the union

of infinite many equivalence classes cannot be defined independently of the chosen representatives!

THEOREM 4.6. Let÷ be a class of torsion groups. Then÷4RC^(R) for some rational group R if and only if the following conditions are satisfied:

(i) ÷ contains all torsion cotorsion groups;

(ii) ÷ is closed under epimorphic images;

(iii) _nv

5

^Z(pn)÷ if and only if ÷ contains all p-groups for all primes p;

(iv) If P is an infinite set of primes then,_p

5

_P^Z(p)÷if and only if _p

5

_P^Tp÷ for all p-groups T_p÷;

(v) If P is an infinite set of primes such that _pP

5

^{Z(p)÷ then}

there exists an infinite subset P8of P such that_p

5

_X^Z(p)÷for all infi- nite X’P8;

(vi) If P is an infinite set of primes such that ÷ contains only bounded p-groups for pP, then _p

5

_P^Z(p)÷;

(vii) The set M÷ contains a unique maximal element.

PROOF. One implication is almost trivial. IfRis a rank-1 group, then obviously conditions (vi) and (vii) are satisfied; for (vii) define P4 ]p P:x_R( 1 )c0( to see that [P] is maximal inM÷. Hence it remains to show the converse implication. As in the proof given in [SW] of Theorem 4.1 define the set

M4

m

^pP

_N

nv

⁵

Z(pⁿ)÷

n

and let S4Q^(M)4

m

^mn :m,nZ,n is a product of powers of primes fromM

n

. Moreover, let [Q] be the maximal element inM÷and putR4 4

o

^{1p: pQ}

p

. Note that the type ofRis independent of the representative chosen for [Q]. By the proof of Theorem 4.1 (see [SW]) all we have to show is that for C4_p

5

_M^Q(p)5_P

5

P

R_P we have RC^(C)₄^RC^(S5R) whereQ^(p)andR_Pare defined as in the proof of Theorem 4.1 (see [SW]).

It is easy to see thatRC^(S)₄^RC

g

_p

⁵

_M^Q(p)

h

using condition (vi). More-

over, sinceQP we clearly haveRC

g

_P

⁵

_P^RP

h

^’RC(R). Conversely, if a torsion group T satisfies Ext (R,T)40 , then T_p40 for almost all primespQ. But [Q] is uniquely maximal inM÷and hence any setPP is almost contained in Qand therefore also T_p40 for almost all primes pP. Thus Ext (R_P,T)40 which shows RC^(R)’RC

g

_P

⁵

_P^RP

h

^{. r}

By Theorem 2.4 any finite rank torsion-free group must satisfy the conditions (i) to (vii) of Theorem 4.6. It is surprising that condition (vi) is easily checked for finite rank torsion-free groups (in fact for countable groupsG condition (vi) is equivalent to saying that OT (G) is Q-closed) but condition (vii) is less obvious and hard to verify.

Note that both conditions (vi) and (vii) may fail in the countable rank case even for completely decomposable groups. (see Example 3.9 and Example 3.10).

Finally we state some properties of the set M÷. LEMMA 4.7. Let G be a torsion-free group. Then

(i) If [P]M^RC(G)is uniquely maximal, then[P]satisfies: If for some set of primes X, XOP is finite, then _p

5

_X^Z(p)RC(G);

(ii) IfMRC(G)is inductive, then there exists a unique maximal el- ement [P] in M^RC(G);

(iii) every chain in M^RC(G) is countable.

PROOF. LetTX4_pX

5

Z(p) and assumeTXRC(G), then there exists by [SW, Proposition 3.5] an infinite subset Y of Xsuch that _p

5

_Y^Z(p)

RC(G) and for every infinite W’Ywe have _pW

5

^Z(p)RC(G). But by the unique maximality of [P] we obtain that Yis almost contained inP, hence XOP is infinite - a contradiction. Thus (i) holds.

The proof of (ii) is trivial: for if [P] and [Q] are distinct maximal ele- ments, then [PNQ] is again an element ofMRC(G)contradicting the max- imality of [P] and [Q].

Finally, if we have a chain ][X_i] :iI( in M^RC(G), then I must be countable since by definition X_i is almost contained in X_i11 for every i and the set P of primes is countable. r

It is not clear for which torsion-free groupsGthe setMRC(G)is induc- tive but it seems unlikely that there is a «more natural» characterization of the groups in R.

R E F E R E N C E S

[A] D. M. ARNOLD,Finite rank torsion-free groups and rings, Lecture Notes in Math. 931, Springer Verlag, Berlin-Heidelberg-New York (1982).

[AD] D. ARNOLD- M. DUGAS,Representation type of finite rank almost com- pletely decomposable groups, Forum Math.,10 (1998), pp. 729-749.

[AV] D. ARNOLD- C. VINSONHALER,Typesets and cotypesets of rank-2 torsion free abelian groups, Pacific J. Math.,114 (1984), pp. 1-21.

[B] R. BAER, The subgroup of the elements of finite order of an abelian group, Ann. of Math., 37 (1936), pp. 766-781.

[EM] P. C. EKLOF- A. MEKLER,Almost Free Modules; Set-Theoretic Methods, North-Holland (1990).

[F1] L. FUCHS, Infinite Abelian Groups I, Academic Press (1970).

[F2] L. FUCHS, Infinite Abelian Groups II, Academic Press (1973).

[F3] L. FUCHS,A survey of Butler Groups of Infinite Rank, Contemp. Math., 171(1994), pp. 121-139.

[G] P. GRIFFITH, A solution to the splitting mixed group problem of Baer, Trans. Amer. Math. Soc., 139(1969), pp. 261-269.

[J] B. JONSSON, On direct decompositions of torsion-free abelian groups, Math. Scand.,5 (1957), pp. 230-235.

[S] L. SALCE,Cotorsion theories for abelian groups, Symposia Mathematica, 23 (1979), pp. 11-32.

[Sch] P. SCHULTZ,The typeset and cotypeset of a rank two abelian group, Pacif- ic J. Math.,70 (1978), pp. 503-517.

[SW] L. STRU¨NGMANN - S. L. WALLUTIS, On the torsion groups in cotorsion classes, to appear in Proc. of the int. Perth Conf. (2000).

[W] R. B. WARFIELD JR., Homomorphisms and duality for torsion-free groups, Math. Z., 107 (1968), pp. 189-200.