Self-<span class="mathjax-formula">$c$</span>-injective abelian groups

Self-c-Injective Abelian Groups (*).

SIMIONBREAZ(**) - GRIGORECAÆLUGAÆREANU(***)

ABSTRACT- Using suitable bibliography, torsion self-c-injective Abelian groups can be characterized, and torsion free self-c-injective Abelian groups turn out to coincide with a widely studied class: the quasi-pure-injective groups. In this paper the mixed case is dealt with: we prove that an honest mixed Abelian group is self-c-injective if and only if it is a direct sum of a divisible group, a quasi-pure- injective torsion free group, and a fully invariant subdirect product of a rigid family of reduced quasi-injective primary groups.

1. Introduction and background.

Recently (2000 and 2004), moduleswhich are self-injective relative to closed submodules were considered by C. Santa-Clara and P.F. Smith (see [13] and [14]) and some Abelian group examples given. The aim of this paper is to determine, up to the infinite rank torsion-free case, the self-c- injective Abelian groups. All the groups we consider in the sequel are Abelian groups. IfAissuch a group,T(A) denotesthe torsion part ofA, T_p(A) isthe p-component ofA,p(A) isthe set of all primesp such that T_p(A)6ˆ0, andd(A) isthe set of all primespsuch thatAisp-divisible. If x2Awe denote byh_p(x) thep-height ofx.Pdenotesthe set of all prime numbers, which we usually denote byporq. A subgroupCAisa closed subgroup if it has no proper essential extensions in A. All unexplained notionsand notationscan be found in [6] and [5].

(*) This work was supported by the Kuwait University Research Administra- tion Grant SM No. 01/04

Mathematics Subject Classification 2000: 20K40, 16D50

(**) Indirizzo dell'A.: ``BabesÎ-Bolyai'' University, Faculty of Mathematics and Computer Science, Str. Mihail KogaÆlniceanu 1, 400084 Cluj-Napoca, Romania;

e-mail address: bodo@math.ubbcluj.ro

(***) Department of Mathematicsand Computer Science Faculty of Science, Kuwait University, Safat 13060 Kuwait; e-mail addres: calu@sci.kuniv.edu.kw

IfP isa property the submodulesof an R-module N may have (e.g., closed, pure, essential, uniform), a module M iscalled N-P-injective if every module homomorphismf :K!Mfrom a submoduleKofNhaving propertyPcan be extended to an endomorphism ofM(i.e., there exists a homomorphismf :N!Msuch thatfj_K ˆf). IfMˆN thenMiscalled quasi-P-injective(orself-P-injective). Therefore

forP closed submodule,N-c-injectiveandself-c-injectivemodules (considered in the above mentioned papers) are defined,

forP pure submodule, N-pure-injective andquasi-pure-injective modulesare defined,

forP any submodule, N-injective and quasi-injective (or self-in- jective) modules are defined.

Similar notions, defined merely for Abelian groups, were intensively studied in the late 70's. In what follows, we recall some of these, since this will be useful in the sequel:

forP p-pure subgroup, the quasi-p-pure-injective groupsare de- fined,

forP neat subgroup, thequasi-neat-injectivegroupsare defined, forP p-neat subgroup, the quasi-p-neat-injectivegroupsare de-

fined.

Here, for a given prime p, a subgroup H of a group G is p-neat if pHˆH\pG,p-pureifpⁿHˆH\pⁿGholdsfor every positive integern and isneat(pure) if it isp-neat (respectivelyp-pure) for all prime numbersp.

As it is well-known, the closed subgroups (Z-submodules) are exactly the neat subgroups ([12] and [11]). Thus, an alternate denomination for self-c-injective Abelian groups (in the spirit of the late 70's) isquasi-neat- injective.

We recall some results which are often used without explicit mention.

LEMMA1.1. If H is a neat fully invariant subgroup of a self-c-injective group G then H is self-c-injective.

LEMMA 1.2. [13, Lemma 2.5]. Let G be a group and let Ai, i2I, be a family of groups. ThenQ

i2IA_iis G -c-injective if and only if each A_iis G-c- injective.

LEMMA1.3. [13, Corollary 2.6]. A direct summand of a self-c-injective group is a self-c-injective group.

Torsion self-c-injective groups.By standard arguments, in the torsion case we can reduce our problem top-groups. Indeed, it is readily checked thatif GˆL

pGpis the primary decomposition of a torsion group, then G is self-c-injective if and only if G_pare self-c-injective for all primes p.

Note that for the proof of this result, the fact that for every subgroupHof Gq we have Hom(H;L

p6ˆqGp)ˆ0 is essential, that is, the fully invariant property of all the primary componentsalone isnot sufficient in order to complete a proof. Actually, as a consequence of Theorem 2.2 we can see that the groupZ^p(Z(p²)Z(p)) isnot self-c-injective but it isa direct sum of two fully invariant subgroups which are self-c-injective (here Z^pˆ fm=p^kjm;k2Zg).

Obviously, a subgroup of a p-group is p-neat if and only if it isneat.

Hence a p-group isquasi-p-neat-injective if and only if it isquasi-neat- injective. Asquasi-neat injectiveself-c-injective, a p-group isself-c-in- jective if and only if it isquasi-p-neat-injective.

In [4], the torsion quasi-p-neat injective groupswere characterized:

THEOREM1.4. A torsion group G is quasi-p-neat-injective if and only if Gpis the direct sumof two quasi-injective groups and Gq is quasi-in- jective for every prime q6ˆp.

In order to have the whole picture, also recall (see [10] and [7]):

THEOREM1.5. The following conditions are equivalent for a group G:

i) G is quasi-injective;

ii) G is a fully invariant subgroup of a divisible (injective) group;

iii) G is either injective or a torsion group whose p-components are direct sums of isomorphic cocyclic groups.

Therefore, the quasi-injectivep-groupsare thehomogeneous p-groups, i.e. direct sums of isomorphic cocyclic p-groups(Z(p^k) with 1k 1), and a self-c-injectivep-group is a direct sum of at most two homogeneous p-groups. We can state a complete characterization for torsion self-c-in- jective groups.

THEOREM1.6. A torsion group is self-c-injective if and only if every primary p -component is a direct sum of at most two homogeneous p- groups.

From Lemma 1.1 and the previous theorem we obtain

COROLLARY1.7. IfGis a self-c-injective group thenT(G)is a self-c- injective group. Consequently, every p-component of Gis a direct sum- mand ofG.

Torsion free self-c-injective groups. For torsion-free groups, neat- ness and purity are equivalent. Hence, the self-c-injective torsion free groups are exactly the quasi-pure-injective torsion free groups.

This class was intensively studied in the last 30 years (though a com- plete characterization does not exist): rank two self-c-injective torsion-free groups were characterized in [1], completely decomposable self-c-injective groupswere characterized in [3] and finally, finite rank self-c-injective torsion free groups were characterized in [2]. Generalizations of this notion were studied in [15], [8] , and [9].

2. Mixed self-c-injective Abelian groups.

According to [6, Exercise 31.17] or [12, Section I.6] every subgroup of a group can be embedded in a neat subgroup which is minimal with respect to this property. For reader's convenience we construct such a subgroup in a particular case.

LEMMA 2.1. Let G be a group and p a prime number. If x2G is an element of infinite order such that hp(x)ˆ0then there exists a neat sub- group[x]of G such that x2[x], and[x]is a rank 1 torsion free group which is not p-divisible.

PROOF. Let Dbe the divisible hull of G. We consider a direct de- composition DˆT(D)K such that x2K. If Uˆ fy2Kj 9n2Z;

(n;p)ˆ1:nk2 hxig \G, we prove that U isa neat subgroup of G. If z2Gissuch thatpz2U, writingzˆtkwitht2T(D) andk2K, we observe thatptˆ0 and hencet2G. Therefore we can supposez2K\G and there exist m;n2Z such that (n;p)ˆ1 and npzˆmx. Since hp(x)ˆ0, p divides m. We obtain inK the equality npzˆm⁰px which yieldsnzˆm⁰x2 hxiand soz2U. HenceUisp-neat inG. Ifq6ˆpisa prime andqz2U(z2G) a similar proof allows us to supposez2K\G.

In thiscontext using the definition of U we obtainz2U. It is not dif- ficult to observe that [x]ˆU verifiesthe required conditions. p Recall that a group isahonest mixed groupif it isneither torsion nor torsion free, and it istorsion reducedif itstorsion part isreduced.

THEOREM2.2. Let G be a torsion reduced honest mixed group which is self-c-injective. Then

a) If p is a prime such that Tp(G)6ˆ0then G=Tp(G)is p-divisible.

b) For every prime p the p-component T_p(G) is a quasi-injective group.

c) There exists a direct decomposition GˆHK where H is a torsion-free group which is p-divisible for all p2p(G)and K is a subdirect product ofQ

pTp(G).

PROOF. a) Since everyp-component ofGisa bounded direct summand, it suffices to prove that if H isnot torsion, H isnot p-divisible and Tp(H)ˆ0, then for every integern>0 the group G⁰ˆZ(pⁿ)Hisnot self-c-injective. SinceHisnotp-divisible, there existsh2Han element of infinite order such that h_p(h)ˆ0. If c be a generator for Z(pⁿ) then xˆc‡pⁿhisof infinite order. Since T_p(H)ˆ0 we have h_p(pⁿh)ˆn. In what follows we use the neat subgroup [x]G⁰.

Let p:G⁰!Hbe the canonical projection. We consider an element u2[x]. Then there exist q;r2Z, (r;p)ˆ1 such that ruˆqx. Thus rp(u)ˆqp(x)ˆqpⁿhand, sinceriscoprime withp, we obtainp(u)2pⁿH.

Moreover, sinceT_p(H)ˆ0, for eachu2U, there exists only one element y_u2Hsuch thatpⁿ(y_u)ˆp(u) , and the correspondenceu7!y_udefinesa homomorphisma:[x]!H. Note thata(x)ˆh.

IfG⁰were self-c-injective there would existb:G⁰!Hwhich extendsa.

Thisleadsto a contradiction since 2nh_p(b(p²ⁿh))ˆh_p(a(pⁿx))ˆ

ˆh_p(pⁿh)ˆn.

b) Assume that for a primepthep-componentT_p(G)6ˆ0 isnot quasi- injective. Then we can find a direct summand ofGisomorphic to a group of the formLˆZ(p^m)Z(pⁿ)Hwith 0<m<n<1, andHa non tor- sion group withTp(H)ˆ0. This group is self-c-injective as a direct sum- mand of a self-c-injective group, hence a) shows that H is p-divisible.

Therefore, every equationp^kyˆh2Hhasa unique solution inH, denoted p ^kh. We consider an elementxˆa‡p^{n m}b‡c whereaisa generator forZ(p^m),bisa generator forZ(pⁿ),c2Han infinite order element and the neat subgroup [x]. Then [x]=pⁿ[x]Z(pⁿ) isgenerated by the coset of xand there exists a homomorphisma:[x]!Lsuch thata(x)ˆb. SinceL is self-c-injective there exists b:L!L extending a. If b(p^{m n}c)ˆ

ˆka‡lb‡d, where k;l are integers, and d2H then 06ˆp^mbˆ a(p^mx)ˆa(p^mc)ˆb(p^mc)ˆpⁿd, a contradiction sincepⁿdisof finite or- der if and only ifdˆ0.

c) For each prime p we consider a direct decompositionGˆT_p(G)

G(p) with the canonical projection pp:G!Tp(G) (note that the de- composition is unique as a consequence of a)), and we view pp asen- domorphisms ofG; actually these are central endomorphisms. The family of allp_pinducesa homomorphismc:G!Q

pT_p(G),c(g)ˆ(p_p(g))_p. Denot- ing byHˆker(c)ˆT

pG(p), we claim thatHisa pure subgroup ofG. To prove this, we consider q2p(G), n>0 an integer and g2G such that qⁿgˆh2H. We writegˆgq‡g(q) withgq2Tq(G) andg(q)2G(q). Using h2G(q) we obtainqⁿgq ˆ0, hence we can supposeg2G(q). Taking an- other primep6ˆqand the decompositiongˆg_p‡g(p) withg_p2T_p(G) and g(p)2G(p), sincehˆqⁿgˆqⁿg_p‡qⁿg(p)2G(p), we obtaing_pˆ0 and so g2G_p. Thereforeg2HandHisq-pure for allq2p(G). Similarly one can see thatHisq-pure for allq2=p(G), henceHisa pure subgroup ofG.

ThenH‡T(G)ˆHT(G) isa pure subgroup ofG(see [6, Exercise 26.5]). SinceG is self-c-injective, it follows that there exists an endomor- phismW:G!G which extendsthe homomorphism 1_H0:HT(G)!

!HT(G). Moreover, sinceW(H)H, we can consider the endomorphism W:G=H!G=H,W(g‡H)ˆW(g)‡H. ThenWinducesthe endomorphismW^? ofc(G) byW^?((pp(g))_p)ˆ(pp(W(g)))_pˆ(W(pp(g)))_pˆ0 for allg2G. Therefore Wˆ0, henceW(G)H, and thisshowsthatWisan idempotent endomorphism ofG. ThenHˆW(G) isa direct summand, andGHc(G). p

LEMMA2.3. Let GˆAB be a group such that:

(1) for every neat subgroup KG, K\Brad_K(A)ˆ\

fKer(f)jf :K!Ag;

and

(2) A is injective relative to the exact sequence0!p_A(K)!A (here pA:G!A denotes the canonical projection).

Then A is G-c-injective.

PROOF. Let K be a neat subgroup of G, and f :K !A a homo- morphism. We denote by r:K!K=K\B the canonical epimorphism.

Since K\BradK(A) we have K\BKer(f), and so there exists a homomorphismf^? :K=K\B!Asuch thatf ˆf^?r. Moreover, there ex- istsp^?:K=K\B!Asuch thatp_AjKˆp^?r. It isnot difficult to prove that p^?isa monomorphism, andp^?(K=K\B)ˆp_A(K). Then by (2) there exists f₁:A!Asuch thatf^?ˆf₁p^?. It followsthatf ˆf^?rˆf₁p^?rˆf₁p_AjK. We view the canonical projectionspA:G!AandpB:G!Basendomorph-

isms of G. If f :G!A isdefined by fpAˆf1, and fpBˆ0 we obtain f_jKˆf(pA‡pB)_jKˆfpAjK ˆf1pAjKˆf, henceAisG-c-injective. p REMARK2.4. GenerallypA(K) isnot a neat subgroup. For example, if AˆZandBˆZ(p) thenKˆ h(p;b1)iisa neat subgroup inGˆAB, but p_A(K)ˆpZisnot a neat subgroup inAˆZ.

COROLLARY2.5. Let GˆAB be a group such that for every neat subgroupKG,K\BradK(A). If

a) A is quasi-injective, or

b) A is a self-c-injective torsion free group and d(A)[d(B)ˆP, then A is G-c-injective.

PROOF. a) is obvious, so we only prove that b) is a consequence of the previouslemma. Letpbe a prime such thatBisp-divisible. IfKisa neat subgroup ofGanda2Aissuch thatpa2p_A(K) then there existk2Kand b2Bsuch that kˆpa‡b. SinceBisp-divisible, pdividesk, and there exists k⁰2K such that pk⁰ˆk. Then there exists a⁰2pA(K) such that paˆpa⁰. Thereforep_A(K) isp-pure inA(Aistorsion free) for all primesp for whichAisnotp-divisible. Thus the pure hull ofp_A(K), asa subgroup of A, is p_A(K)_?ˆZ_Sp_A(K)ˆ f(m=n)xjm=n2Z_S; x2p_A(K)g, where S is the set of all primespsuch thatAisp-divisible, andZ_Sˆ fm=nj(n;p)ˆ1 for all p2=Sg. Ifa:pA(K)!Aisa homomorphism then a^\:pA(K)_?!A, a^\((m=n)x)ˆ(m=n)a(x) iswell defined and extendsa. SinceAisself-c-in- jective we can extenda^\ to a homomorphisma^?:A!A, and the proof is

complete. p

LEMMA 2.6. If A is a reduced quasi-injective p-group, and B is a p- divisible group then A is AB-c-injective.

PROOF. Letn1 (the casenˆ0 isobvious) be the exponent of thep- groupA(i.e.AL

Z(pⁿ)),Ka neat subgroup ofGˆABanda:K!A a homomorphism. Ifl2K\B, thenlisdivisible bypinGand there exists l12Ksuch thatpl1ˆl. IfpA:G!Aisthe canonical projection, we have ppA(l1)ˆ0. In the hypothesis n>1, l1 isdivisible by p, hence we can continue thisprocedure in order to construct l22K such that pl2ˆl1. Continuing in the same way we obtain a sequencel₀ˆl;l₁;. . .;l_ninKsuch that for all i2 f1;. . .;ng we have pl_iˆl_{i 1}. Then K\BpⁿK and K\Brad_A(K) followsand finally we use Corollary 2.5 in order to con-

clude thatAisAB-c-injective p

The next result is a generalization of [14, Theorem 6].

COROLLARY2.7. LetSbe an infinite set of primes. If for each prime p2S T_p denotes a non-zero reduced p-group, then the group Q

p2ST_p is self-c-injective if and only if everyT_p is a quasi-injective group.

PROOF. Using Lemma 1.1 it suffices to prove that eachTqisQ

p2STp-c- injective, and this is a consequence of Lemma 2.6. Conversely, the result followsfrom Theorem 2.2 sinceSisan infinite set. p We are now able to characterize the direct summandsKwhich appear in Theorem 2.2, c).

THEOREM2.8. Let G be a group such that GQ

pTp(G), and each p- component Tp(G)is a reduced quasi-injective group. The following con- ditions are equivalent:

i) G is a self-c-injective group;

ii) G is a fully invariant pure subgroup ofQ

pT_p(G);

iii) G is a fully invariant subgroup ofQ

pTp(G).

PROOF. i))ii) Using Theorem 2.2 a),G=Tp(G) is p-divisible for all p2p(G) and soG=T(G) isp-divisible for allp2p(G). Hence it isp-pure in Q

pT_p(G)=T(G), and it isnot difficult to prove thatGisp-pure inQ

pT_p(G) for allp2p(G).

SinceG=T(G) isp-divisible for allp2p(G) we have Hom(G=T(G);G)Hom(G=T(G);Y

p

Tp(G))ˆ0:

Therefore the exact sequence

0ˆHom(G=T(G);G)!Hom(G;G)!Hom(T(G);G)

provesthat two endomorphisma;bofGcoincide if and only ifa_jT(G)ˆb_jT(G). Note thatQ

pTp(G) hasthe same property.

Letq2=p(G). The homomorphisma:T(G)!T(G) given byqa(x)ˆxis well defined. SinceG is self-c-injective there existsb:G!G which ex- tends a. Then qb_jT(G)ˆ1_jT(G) and, using the previous paragraph, this provesthat the multiplication byqisan automorphism ofG. ThereforeGis q-divisible, and soq-pure inQ

pTp(G).

In order to prove thatGis fully invariant we consider an endomorphism a:Q

pTp(G)!Q

pTp(G). Then there exists b:G!G which extends

a_jT(G). Moreover by Corollary 2.7,Q

pTp(G) isself-c-injective andGispure inQ

pTp(G), hence there existsg:Q

pTp(G)!Q

pTp(G) which extendsb.

Sincea_jT(G)ˆg_jT(G) we obtainaˆg, and soa(G)G.

The implication ii))iii) isobvious. Asfor iii))ii) observe that for every q2=p(G) the homomorphisma:Q

pT_p(G)!Q

pT_p(G) given byqa(x)ˆxis well defined. Sincea(G)Gwe obtainqGˆG, so thatGisq-divisible. If q2p(G) we consider a:Q

pTp(G)!Q

pTp(G) defined by a(Tq(G))ˆ0 and qa(x)ˆx for all x2Q

p6ˆqTp(G) and the direct decomposition GˆT_q(G)(G\Q

p6ˆqT_p(G)). It followsthat a(G\Q

p6ˆqT_p(G))G\

\Q

p6ˆqT_p(G), and thisshowsthat G\Q

p6ˆqT_p(G) is q-divisible. Then G=T(G) isq-pure in (Q

pTp(G))=T(G), henceGisq-pure inQ

pTp(G).

ii))i) isa consequence of Corollary 2.7 and Lemma 1.1. p An honest mixed groupGwhich verifiesthe conditionsin the previous theorem iscalled aproper mixed self-c-injectivegroup.

COROLLARY 2.9. If G is a proper mixed self-c-injective group then G=T(G)is divisible andGisp-divisible wheneverT_p(G)ˆ0.

Recall that a groupGisasplittinggroup ifT(G) isa direct summand ofG.

COROLLARY 2.10. A self-c-injective group G with r₀(G) @₀ is a splitting group.

PROOF. Using Theorem 2.2, if a self-c-injective group is not splitting, it has a direct summand which is a proper mixed self-c-injective group.

Therefore, it suffices to prove that a proper mixed self-c-injective group has the torsion free rank at least 2^@0.

LetGbe a proper mixed self-c-injective group. We consider the group HˆQ

pTp(G) together with the canonical projectionspp:H!Tp(G). For an elementx2Hwe consider thecharacteristicx(x)ˆ(hp(pp(x)))_p. Since eachT_p(G) isa homogeneousboundedp-group, there existsC_p(x) a cyclic direct summand of T_p(G) which contains p_p(x). Therefore x2Q

pC_p(x) which isa direct summand of H. Thus, for two elements x;y2Hthere exists an endomorphism ofHwhich mapsxintoyif and only ifx(x)x(y).

Therefore ifx2Gisan infinite order element,x(x) has@0finite com- ponents. Ifxˆ(x_p)_pis a characteristic, for each setSPwe consider the characteristicx^Sˆ(x^S_p)_pwithx^S_pˆ 1for allp2Sandx^S_pˆx_pfor allp2=S.

If xˆx(x), then x^S isthe characteristic of x^S2T(Q

pC_p(x)) with

pp(x^S)ˆ0 for allp2S, andpp(x^S)ˆpp(x) for allp2=S. It followsthat the cardinality of G\Q

pCp(x) is 2^@0, and since T(Q

pCp(x)) isa countable group, the cardinality of (G\Q

pCp(x))=T(G\Q

pCp(x)) is2^@0, and thisis possible if and only if the torsion free rank ofGisat least 2^@0. p

THEOREM2.11. Let G be an honest mixed group. Then G is self-c-in- jective if and only if GˆDHR such that

i) D is a divisible group,

ii) R is a self-c-injective reduced torsion free group which is p-divi- sible for all p2p(H), and

iii) H is a proper mixed self-c-injective group or it is a quasi-injective reduced torsion group.

PROOF. If Gisself-c-injective then G hasthe mentioned form (con- sequence of Theorem 2.2). IfGhasthisform thenGisa pure fully invariant subgroup of G⁰ˆDR(Q

pT_p(H)). Therefore, in order to prove the converse it suffices to show thatG⁰isself-c-injective.

SinceDisdivisible,DisG⁰-c-injective. LetKbe a neat subgroup ofG⁰, andf :K!R a homomorphism. If q isa prime such thatq2=p(H) then DHisq-divisible. Letx2K\(DH). Thenq dividesxinG⁰, and so there exists y2K such that qyˆx. Writing yˆr‡z, with r2R and z2DH , we obtainqrˆ0, hencerˆ0, since R istorsion free. Then y2K\(DH) and it followsthat f(K\(DH)) is q-divisible for all q2=p(H). Thereforef(K\(DH)) iscontained inT

q2=p(H)q^vR. However, sinceRisp-divisible for allp2p(H) we obtainT

q2p(H)= q^vRasthe divisible part of the reduced group R. Hence, f(K\(DR))ˆ0, and R isG⁰-c- injective asa consequence of Theorem 2.2, Theorem 2.8, and Corollary 2.5.

SinceT_p(H) isG⁰-c-injective by Lemma 2.6, an application of Lemma 1.1

completesthe proof. p

REFERENCES

[1] D. ARNOLD- C. I. VINSONHALER- W. J. WICKLESS,Quasi-pure projective and injective torsion free abelian groups of rank 2. Rocky Mountain J. Math. 6, no.

1 (1976), pp. 61-70.

[2] D. M. ARNOLD- B. O'BRIEN- J. D. REID,Quasipure injective and projective torsion-free abelian groups of finite rank. Proc. London Math. Soc. (3) 38, no.

3 (1979), pp. 532-544.

[3] K. BENABDALLAH- A. LAROCHE, Les groupes abeÂliens quasi-purs injectifs sans torsion compleÁtement deÂcomposables. (French) Rend. Mat. (6) 10 no. 2-3 (1977), pp. 305-312.

[4] K. BENABDALLAH- R. BRADLEY- A. LAROCHE,Sur les groupes quasi-p-nets injectifs et projectifs. (French) Abelian group theory (Proc. Second New Mexico State Univ. Conf., LasCruces, N.M., 1976), Lecture Notesin Math., Vol. 616, Springer, Berlin, 1977, pp. 228-239.

[5] N.V. DUNG- D.VANHUYNH- P.F. SMITH- R. WISBAUER,Extending modules.

Harlow (1994).

[6] L. FUCHS,Infinite Abelian groups, vol. 1 (1970) and 2 (1973), Academic Press.

[7] L. FUCHS- L. SALCE, Modules over non-Noetherian domains. Mathema- tical Surveysand Monographs, 84. American Mathematical Society, Provi- dence, RI (2001).

[8] P. GOETERS- C. VINSONHALER- W. WICKLESS,A generalization of quasi- pure injective torsion-free abelian groups. Houston J. Math. vol. 22, no. 3 (1996), pp. 473-384.

[9] H.P. GOETERS- W.J. WICKLESS,Qqpi groups and quasi-equivalence. Proc.

AMS, vol. 126, no. 11 (1998), pp. 3145-3150.

[10] M.A. KIL'P,Quasi-injective abelian groups.[Russian], Vestnik Mosk. Univ., 22 nr. 3 (1967), pp. 3-4.

[11] F. LOONSTRA,Closed submodules. Abelian group theory (Honolulu, Hawaii, 1983), Lecture Notesin Math., 1006, Springer, Berlin-New York, (1983), pp.

630-638.

[12] G. RENAULT, EÂtude des sous-modules compleÂments dans un module.

(French) Bull. Soc. Math. France MeÂm. 9, (1967) 79 pp. ???

[13] C. SANTA-CLARA- P.F. SMITH, Modules which are self-injective relative to closed submodules. In 'Algebra and its Applications' (Athens, Ohio, 1999), Contemporary Mathematics259, AMS, Providence, (2000), pp. 487- 499.

[14] C. SANTA-CLARA- F. SMITHPATRICK,Direct products of simple modules over Dedekind domains. Arch. Math. (Basel) 82 no. 1 (2004), pp. 8-12.

[15] C. VINSONHALER,Almost quasi-pure injective abelian groups, Rocky Mount.

J. Math., vol. 9, no. 3 (1979), pp. 569-576.

Manoscritto pervenuto in redazione l'11 luglio 2005.