Localizations of tensor products

Localizations of tensor products

MANFREDDUGAS(*) - KELLYACEVES(*) - BRADLEYWAGNER(*)

ABSTRACT- A homomorphisml:A!BbetweenR-modules is called a localization if for allW2HomR(A;B) there is a uniquec2HomR(B;B) such thatWˆcl. We investigate localizations of tensor products of torsion-free abelian groups. For example, we show that the natural multiplication mapm:RR!Ris a local- ization if and only ifRis an E-ring.

KEYWORDS. Torsion-free abelian groups, tensor products, localizations.

MATHEMATICSSUBJECTCLASSIFICATION(1991). Primary 20K30, 15A15; Secondary 46M05.

1. Introduction

LetRdenote some ring. Many notions in the theory ofR-modules can be stated in terms of some universal property. Here are some examples:

(1) TheR-moduleFisfreewith basisBifBis a subset ofFsuch that for any R-moduleXand any functionf :B!Xthere exists a unique homo- morphismW:F!Xsuch thatWj³³_Bˆf.

(2) TheR-moduleGissmall, if for each familyfX_i:i2IgofR-modules theabeliangroupHom_Rÿ

G;L

i2IX_i

isnaturallyisomorphictoL

i2IHom_R(G;X_i).

(3) TheR-moduleGisstrongly slender[9] if for each familyfX_i:i2Ig ofR-modules the abelian groupHomRÿ Q

i2IXi;G

is naturally isomorphic to L

i2IHomR(Xi;G).

(*) Indirizzo degli A.: Department of Mathematics, Baylor University, Waco, Texas 76798, USA.

E-mail: [email protected] [email protected] [email protected]

(4) Let l2HomR(A;B) be a homomorphism. We call l a split homomorphism if for any R-module X and any W2HomR(A;X) there exists some c2Hom_R(B;X) such that Wˆcl. If one considers the case of XˆA and Wˆid_A, then it is easy to see that l is indeed a splitting homomorphism, i.e.lis injective andl(A) is a direct summand of B.

(5) TheR-moduleG isinjectiveif for any R-module X and any sub- module K of X and any W2Hom_R(K;G) there exists some c2 Hom_R(X;G) such that Wˆcj³³_K. Note that ``projective'' is the dual of

``injective''.

(6) Tensor products: To simplify notation, let us assume RˆZ. Let A;BandTbe abelian groups andt:AB!Ta bilinear map. The pair (T;t) is atensor productofAandB, if for any abelian groupX and any bilinear map s:AB!X, there exists a unique c2Hom(T;X) such thatsˆct. It is well known, of course, that tensor products exist and are unique up to isomorphism.

It would be easy to continue this list. All these definitions can be modified by restricting theX's. Let us do this. We get

(1^s) In (1), replace ``X'' by ``F''. A moduleFsatisfying (1^s) is calledself- freewith basisB, c.f. [5].

(2^s) In (2), replace all the ``X<sub>i</sub>'' by ``G''. Such a moduleGis calledself- small. This notion has been studied by many authors, c.f. [3] and the literature referenced there.

(3^s) In (3), replace all the ``X<sub>i</sub>'' by ``G''. Such a module is calledstrongly self-slender, c.f. [9].

(4^s) In (4) replace ``X'' by ``B'' and add the condition that the mapcis unique. Such a homomorphism l:A!B is called a localization of A.

There is a large amount of literature on localizations. See for instance [6]

and [7] and the papers referenced there.

(5^s) In (5), replace ``X'' by ``G''. Then G is called quasi-injective. If RˆZandKis restricted top-pure subgroups ofG, thenGis calledquasi- p-pure-injective, or qppi for short. Again, there is a lot of literature on this topic dating back to the 1970's, c.f. [1] or [13].

In this paper we focus on a specification (6^s) of (6):

DEFINITION1. Let R be a ring, AˆAR a right R-module, BˆRB a left R-module and T some Abelian group. Following[12, page 207], we call a map t:AB!T a middle linear map if t is bilinear and t(ar;b)ˆt(a;rb)for all a2A;b2B and r2R. Let Midlin_R(A;B;T)de- note the set of all middle linear maps from AB into T. We call the pair (T;t) a qutensor productof A;B over R if for all s2MidlinR(A;B;T), there exists auniquehomomorphismc2HomZ(T;T)such thatsˆct.

The map:AB!A_RB is a middle linear map with(a;b)ˆab for all a2A;b2B.

It follows from the definitions that qutensor products are a combination of the notions in (4^s) and (6):

The pair (T;t) is a qutensor product of the modulesA;Bif and only if there exists a localization l:A_RB!T such that tˆl , where :AB!A_RBis the natural map with(a;b)ˆabfor alla2A andb2B.

In this paper we concern ourselves with localizations of tensor products of torsion-free abelian groupsA;B.

After some preliminaries in Section 2, we study localizations of ar- bitrary direct sums of modules in Section 3. In Section 4, we look at surjective localizations and find conditions for p-reduced torsion-free abelian groups having ap-reduced tensor product. In Section 5, we use the absolute E-rings constructed in [10] and [11] to obtain absolute localizations of torsion-free abelian groups that are p-reduced for in- finitely many primes p. In Section 6, we present some examples of some surprising properties of tensor products of reduced torsion-free abelian groups. For example, there exists a strongly indecomposable reduced abelian group G such that GG is reduced and completely decomposable. In Section 7, we consider torsion-free abelian groups of finite rank whosep-rank is less than their rank. We find an example of such a groupGof rank 4 and p-rank 2 such thatGGis not reduced, but G has no pure subgroups of rank at least 2 but of p-rank 1. (It is well known that GGis notp-reduced if Ghas rank at least 2 andp- rank 1.) We also find a quasi-isomorphism invariant for such groups. In the last section, we take a glimpse at zero product determinated al- gebras, c.f. [4]. Let 12Rbe a ring. Then there exists an epimorphism m:R_ZR!Rwith m(ab)ˆabfor alla;b2R. We show thatmis a localization if and only if R is an E-ring.

2. Definitions and First Results Recall the following, well established

DEFINITION 2. Let M;L;X be modules over some ring and l2 Hom(M;L). Then l?X provided that for any a2Hom(M;X) there is auniqueb2Hom(L;X)such thataˆbl. Ifl?L, thenlis called a localization of M. Abusing notations, sometimes L is called a local- ization of M.

Next we show that qutensors are exactly the localizations of the or- dinary tensor product of the modules.

PROPOSITION1. Let A_Rand_RB be modules andt2Midlin_R(A;B;D).

Then t factors through , i.e. there exists l:ARB!D such that tˆl . Moreover, (D;t) is a qutensor product of A;B if and only if l:A_RB!D is a localization of A_RB.

PROOF. By the universal property of the tensor product, there is a uniquel2Hom(A_RB;D) such thattˆl . Assume thatlis a local- ization and lets:AB!Dbe a bilinear map. By the universal property of the tensor product, there exists a unique mapd2Hom(ARB;D) such thatsˆd . Then there is a uniqueb2End(D) such thatdˆbland we have thatsˆbl ˆbt. On the other hand, assume thatsˆgt.

We infer that gtˆgl ˆbl and thus glˆbl. We con- clude thatgˆband we have that (D;t) is a qutensor ofA;B.

For the other direction, leta2Hom(ARB;D) and putsˆa , a middle linear map from AB into D. Thus there exists a unique b2End(D) such thatsˆbt and thusa ˆbl , which implies that aˆbl. If b⁰2End(D) is some other map with aˆb⁰l, then sˆa ˆb⁰l ˆbl and we infer btˆb⁰t and thus bˆb⁰. This shows thatlis a localization. p

3. Localizations of Direct Sums of Modules We begin with:

PROPOSITION 2. Let l:AˆL

i2IA_i!D be a localization, i.e. l?D.

Then there exists a setfg_i:i2Ig of orthogonal idempotents in End(D) such that:

(a) For Diˆg_i(D) we have that D⁰ˆL

i2IDiD and diˆlj³³_Ai2 Hom(Ai;Di).

(b) d_j?D_ifor all i;j2I:

(c) ker (l)ˆL

i2I(A_i\ker (l)).

(d) If the index set I is finite, then DˆD⁰.

PROOF. Define l_i:A!D by l_ij³³_Aiˆlj³³_Ai and l_i(A_j)ˆ f0g for all i6ˆj2I. Then there exist uniqueg_i2End(D) such that

liˆg_ilfor alli2I.

Let aˆP

j2Ia_j2A where a_j2A_j. Then (g_il_i)(a)ˆ(g_il_i)(a_i)ˆ (g_il)(a_i)ˆl_i(a_i)ˆl_i(a) and we have:

l_iˆg_il_ifor alli2I.

Now we have (g²_i)lˆg_i(g_il)ˆg_il_iˆl_iˆg_il and we infer thatg_iis an idempotent element ofEnd(D).

Let i6ˆj2I and a2A as above. Then (g_ig_jl)(a)ˆ(g_il_j)(a)ˆ g_i(lj(aj))ˆg_i(l(aj))ˆli(aj)ˆ0. This shows that (g_ig_j)lˆ0ˆ0l and it follows thatg_ig_jˆ0 for alli6ˆj2I. We infer thatfg_i:i2Igis a set of orthogonal idempotents and thus D⁰ˆL

i2ID_i is a submodule of D whereD_iˆg_i(D).

Let a_i2A_i. Then l(a_i)ˆl_i(a_i)ˆg_i(l_i(a_i))g_i(D)ˆD_i. This shows thatdi2Hom(Ai;Di).

Note thatDˆD_iker (g_i) andg_j(D)ker (g_i) for alli6ˆj2I.

To show thatd_iˆlj³³_Ai:A_i!D_iis orthogonal toD_j, leth2Hom(A_i;D_j) for somej2I. Thenhnaturally extends to a maph⁰fromAtoDby setting the maph⁰equal to 0 on the other summands ofA. Then there exists a unique mapu:D!Dsuch thath⁰ˆul. Letaˆai‡a⁽ⁱ⁾ 2Awhereai2Aiand a⁽ⁱ⁾2 L

i6ˆj2IAj. Then (ul)(a)ˆ(ul)(ai)‡(ul)(a⁽ⁱ⁾)ˆh⁰(a)ˆh(ai)2Dj. It follows that (ul)(a⁽ⁱ⁾)ˆ0 by settingaiˆ0. Sinceg_jacts as the identity map onD_j, we get (g_juj³³_Di)(lj³³_Ai)ˆh. Letu_iˆg_juj³³_Di 2Hom(D_i;D_j).

Thenhˆu_id_i.

Next we need to show that the mapu_iis unique with that property. Let u⁰_ibe another such map, and note that l(A⁽ⁱ⁾)ker (g_i). Letu⁰2End(D) with u⁰j³³_Diˆu⁰_i and u⁰(ker (g_i))ˆ f0g. It follows that h⁰ˆu⁰l and since l?D, we infer thatuˆu⁰and thusu_iˆu⁰_i.

Let aˆP

i a_i2ker (l) where a_i2A_i. Then 0ˆP

il(a_i)ˆP

il_i(a_i)ˆ P

i g_i(l_i(a_i))2L

i2ID_i and it follows thatl(a_i)ˆ0 for all i2I. This shows that ker (l)L

i2I(A_i\ker (l))ker (l) and (c) follows.

Now assume thatIis finite. ThenidDlˆP

i2IliˆP

i2Ig_ilˆ P

i2Ig_i

l and it follows thatid_DˆP

i2Ig_i. p

COROLLARY1. Given abelian groups D and AˆL

i2IA_i with I finite.

Letl2Hom(A;D). Ifl?D, then DˆL

i2ID_iandd_iˆlj³³_Ai2Hom(A_i;D_i) satisfiesd_i?D_j for all i;j2I.

Conversely, if DˆL

i2ID_i and l_i2Hom(A_i;D_i) with l_i?D_j for all i;j2I, thenlˆL

i2Ili?D.

COROLLARY2. Let AˆL

i2IA_iwith finite index set I and K a subgroup of A. The canonical mapp:A!A=K is a localization if and only if

(1)The canonical mapsp_i:A_i!A_i=(K\A_i)are localizations for all i2I and

(2)K ˆL

i2I(K\A_i)andp_i?(A_j=(K\A_j))for all i;j2I.

PROOF. It follows from Proposition 2 that (1) and (2) are necessary.

Suppose (1) and (2) hold. Let W2Hom(A;A=K) and note that A=K L

i2I(A_i=(A_i\K)) because of (2). There existW_ji:A_i!A_j=(A_i\K) such that W is represented by the matrix [W_ji]]. For each W_ji there is a unique c_ji:A_i=(A_i\K)!A_j=(A_j\K) such thatW_jiˆc_jip_i. It follows from the definitions thatcˆ[c_ji]2End(A=K) is the desired map. p COROLLARY 3. With the above notation, if A_iA_j then D_iD_j for any i;j2I.

PROOF. Let sji:Ai!Aj be an isomorphism with sij:Aj !Ai the inverse map. Extend s_ji to s⁰_ji:A!A by setting s⁰_ji(Aa)ˆ f0g for all i6ˆa2I. Thenls⁰_ji2Hom(A;D) and there exists a uniqued_ji2End(D) such that

(1)ls⁰_jiˆd_jil. In a similar fashion we get (2)ls⁰_ij0 ˆd_ijl:

It follows that g_jlˆl_jˆ(ls⁰_ji)s⁰_ijˆ(d_jil)s⁰_ijˆd_ji(ls⁰_ij)ˆ (d_jid_ij)l. We infer

(3)g_jˆdjidijas well asg_iˆdijdji. It follows that

(4)g_id_ijˆd_ijd_jid_ijˆd_ijg_jandg_jd_jiˆd_jid_ijd_jiˆd_jig_i.

Taking the restrictions toDj,Diwe get

(5)d_ijj³³_Dj ˆg_id_ijj³³_Dj :D_j!D_ias well asd_jij³³_Diˆg_jd_jij³³_Di:D_i!D_j. The equations (3) now imply that dijj³³_Djanddjij³³_Di are a pair of inverse

isomorphisms and we haveD_iD_j. p

PROPOSITION 3. Same notations as above. If D is slender, then DˆL

i2ID_i.

PROOF. Since l is a localization, we have that Q

i2I Hom(Ai;D) HomÿL

i2IAi;D

ˆHom(A;D)End(D). The latter isomorphism:End(D)! Hom(A;D) is given byWˆWlfor allW2End(D). Let^#denote the inverse of.

Pick an elementd2D.

Define a mapsd:End(D)!Dbysd(W)ˆW(d) for allW2End(D). This gives rise to a homomorphismtd:Q

i2IHom(Ai;D)!D. SinceDis slender, there exists a cofinite subset J ofI such that t_dÿ Q

i2JHom(A_i;D)

ˆ f0g.

Note that (li)^#ˆg_ifor alli2Iandtd(lj)ˆ0 for allj2J. But now we have 0ˆtd(lj)ˆg_j(d) for allj2J. We infer thatP

i2Ig_i(d) is a finite sum for all d2Dand thusgˆP

i2Ig_i2End(D) withglˆlˆidDl. It follows that gˆid_Dand thusDˆL

i2ID_i. p

DEFINITION3. Let(A;D)be a pair of abelian groups. We call this pair relatively semi-rigid,if each06ˆW2Hom(A;D)is injective.

For example, if AZ and D is torsion-free, then the pair (A;D) is relatively semi-rigid.

PROPOSITION4. Let I be an index set and Ai, Dibe abelian groups for all i2I. Moreover, letl_i2Hom(A_i;D_i)for all i2I such that:

(1) li?Djfor all i;j2I.

(2) For all i2I, we have that the pair(A_i;D_j)is relatively semi-rigid for all but finitely many j2I.

ThenlˆL

i2Ili:AˆL

i2IAi!DˆL

i2IDiis a localization, i.e.l?D.

PROOF. Let W2Hom(A;D). Then Wˆ[W_ji] where W_ji2Hom(Ai;Dj) and for alli2Iandai2Aiwe haveW_ji(ai)ˆ0 for all but finitely manyj2I.

By (2) we infer thatW_jiˆ0 for all but finitely manyj2I. By (1), there exist uniqueg_ji2Hom(D_i;D_j) such thatW_jiˆg_jil_i. Note thatg_jiˆ0 whenever W_jiˆ0. This implies thatgˆ[g_ji]2End(D) andWˆgl. The uniqueness

ofgwith that property follows immediately. p

COROLLARY4. Let I be an index set and A_i, D_ibe abelian groups for all i2I. Moreover, letl_i2Hom(A_i;D_i)for all i2I such that:

(1)l_i?D_j for all i;j2I.

(2)For all i2I, we have that Hom(A_i;D_j)ˆ0for all but finitely many j2I.

ThenlˆL

i2Il_i:AˆL

i2IA_i!DˆL

i2ID_iis a localization, i.e.l?D.

4. Surjective Localization

PROPOSITION5. Let B be a subgroup of the abelian group A such that:

(1) B is fully invariant in A.

(2) The natural map End(A)!Hom(A;A=B)is surjective.

Then the canonical mapp:A!A=B is a localization.

PROOF. LetW:A!A=Bbe a homomorphism. By (2), there is some u2End(A) such that Wˆpu. Define c:A=B!A=B by c(a‡B)ˆ u(a)‡B. Note that cis well defined by (1) and it follows thatWˆcp.

Sincepis surjective, we infer thatcis unique with that property. p Let Gbe a torsion-free abelian group. For a2G, let k ka denote the type of a2G and G(t)ˆ fg2G:k k g tg. A type tˆ(tp)_p2P is called idempotent (or ring) type, if t_p2 f0;1g for all primes p2P. Moreover

G

k kdenotes the set of all types of the elements ofG.

PROPOSITION 6. Let G be a torsion-free abelian group. Suppose the typetis an element in Ak kand of ring type, i.e. fortˆ(t_p)_p2Pwe have t_p2 f0;1g. Then (G=G(t))(t)ˆ f0g and for all W2Hom(G;G=G(t)) we have G(t)ker (W). Moreover, the natural map p:G!G=G(t) is a localization.

PROOF. Letx2Gsuch thatkx‡G(t)k t. Letpbe a prime such that tpˆ 1. We have that for all n2N, there is some xn2G such that pⁿxnÿxˆkn 2G(t). Nowkn isp-divisible inGand it follows thatxisp- divisible inG. This implies thatx2G(t) and thusx‡G(t)ˆ0. Note that W(G(t))(G=G(t))(t)ˆ f0g. Thus W induces a map eW2End(G=G(t)) by eW(x‡G(t))ˆW(x). It follows that WˆeWp and eW is unique with this property becausepis surjective. This shows thatpis a localization. p

Letpbe a prime integer. ThenZpˆz

n:z2Z;n2N;gcd(p;n)ˆ1 denotes the ring of integers localized atp. The torsion-free abelian group is calledp-locally free ifG_ZZ_p is a freeZ_p-module.

REMARK 1. Let A⁰;B⁰ be pure subgroups of the torsion-free abelian groups A;B respectively. Then A⁰B⁰is a pure subgroup of AB.

To see this, note that 0!A⁰B!AB!(A=A⁰)B!0 is pure- exact, and 0!A⁰B⁰!A⁰B!A⁰(B=B⁰)!0 is pure-exact and purity is transitive. This shows:

PROPOSITION7. Let A;B be torsion-free abelian groups. Then AB is p-reduced if and only if A⁰B⁰ is p-reduced for all pure, finite rank subgroups A⁰;B⁰of A;B respectively.

THEOREM1. Let A;B be torsion-free, p-reduced abelian groups such that all pure, finite rank subgroups of A are p-locally free. Then AB is p-reduced.

PROOF. If there exists a non-zero element win AB of infinitep- height, then there exists a pure, finite subgroupA⁰ofAsuch thatwis an element ofA⁰B. Now consider the localization (A⁰B)_pofA⁰Bat the primep. We have (A⁰B)_pA⁰_pBpP

k BpasZp-modules sinceA⁰_pis a freeZp-module of some rankk. SinceBisp-reduced, theZp-moduleBphas no elements of infinitep-height, which shows that no such elementwexists.

p By an assertion of Warfield's, a torsion-free groupGof finite rankmis p-locally free if and only if mˆrp(G):ˆdim_Z=pZ(G=pG). For the con- venience of the reader, here is an outline of the proof:

Letfai‡pG:1imgbe a basis ofG=pGandBˆ P

1imaiZ. Then fa_i:1imgisp-independent andBis ap-basic subgroup ofG, i.e.Bis a free,p-pure subgroup of GandG=Bisp-divisible and a torsion group with (G=B)[p]ˆ f0g:It follows thatGZpˆBZpis a freeZp-module.

COROLLARY5. Let A;B be torsion-free, p-reduced abelian groups such that for all pure, finite rank subgroups A⁰of A the rank of A⁰is equal to the p-rank rp(A⁰)of A⁰. Then AB is p-reduced.

5. Absolute Localizations

LetAbe some algebraic structure that has some propertyP. ThenA has propertyPabsolutelyifAhas propertyPin any generic extension of the set-theoretic universe in whichAwas originally constructed. Letk(v) denote the first v-ErdoÈs cardinal. In a remarkable paper [10], GoÈbel, Herden and Shelah constructed absolute E-ringsRof cardinalitylfor any infinite cardinal l<k(v). Inspecting their proof, one realizes that the following result was shown:

THEOREM2. [10] Let l<k(v) be a cardinal and Z[X](resp. Q[X]) the polynomial ring in l commuting variable over Z (resp. Q). Then there exists a countable familyfL_i:i<vgof ideals of Z[X]such that

(1)Each Liis a direct summand of the abelian groupZ[X]and (2)fW2EndQ(Q[X]):W(QL_i)QL_ifor all i<vg ˆQ[X]absolutely.

This version will appear in [11].

We will use this result to construct absolute localizations. First we show:

LEMMA1. Let A be a commutativeQ-algebra andFa family of ideals of A such thatfW2EndQ(A):W(J)J for all J2 Fg ˆA. Let V be aQ-vector space. Then fW2Hom_Q(A;V_QA):W(J)V_QJ for all J2 Fg ˆ Hom_A(A;V_QA).

PROOF. Let B be a basis of the vector spaceV. Let pb:VQA! bA be the natural projection with p_b(cA)ˆ f0g for all b6ˆc2B.

Let W2 fW2Hom_Q(A;V_QA):W(J)V_QJ for all J2 Fg. Then p_bW:A!bAA and (p_bW)(J)p_b(V_QJ)ˆbJJ. Thus,

by hypothesis, (pbW)(x)ˆ((pbW)(1))x for all x2A. It follows that W(x)ˆ P

b2Bpb(W(x))ˆP

b2B((pbW)(1))xˆW(1)x and W is A-linear. p LetGbeatorsion-free abeliangroupandPˆ fp_i:i<vganinfinite set of prime integers such thatGisp-reduced for allp2P. Lett_idenote the type of the subring Z1

p_i

of Q. With the notations of Theorem 2, let Rˆ Z[X]‡P

i<vZ1 p_i

L_iQ[X]. ThenRis the absolute E-ring constructed in [10]. Note thatR=Z[X] is a torsion abelian group. By (1) we haveZ[X]ˆ LiTi as abelian groups. Then R=Z[X]ˆP

i<v(Z[X]‡Z1 p_i

Li)=Z[X]ˆ P

i<v

ÿT_iZ1 p_i

L_i

=(T_iL_i)P

i<v

ÿZ1 p_i

L_i

=L_iwhere thei-th summand is a divisiblep_i-group. This shows thatR(t_i)ˆZ1

pi

L_ifor alli<v. Now letW2EndZ(R). ThenW…Z1

p_i L_i

ˆW(R(t_i))R(t_i)ˆZ 1 p_i

L_i. Letc2 End_Q(Q[X]) be the unique homomorphism induced byW. Thenc(QL_i) QL_ifor alli<vand Theorem 2 supplies somed2Q[X] withcˆd. Since dˆc(1)ˆW(1)2R, we haveWˆdfor somed2R, i.e.Ris an E-ring.

Let us call such a ringRan (GHS)-E-ring with prime number setP.

LEMMA2. Let B be a torsion-free abelian group and R,t_ias above. If B(t_i)ˆ f0g, then(BR)(t_i)ˆBR(t_i). (Here ``'' is understood to mean

``_Z''.)

PROOF. Consider the short exact sequence 0!Z[X]!R! R=Z[X]L

i<v

ÿZ1 p_i

Li

=Li

!0, which gives rise to the sequence 0!BZ[X]!BR!L

i<vBÿ Z1

p_i Li

=Li

!0. Note thatZ[X]

is a free abelian group and thusBZ[X] is isomorphic to a direct sum of copies of B and thus has no elements of infinite pi-height. Let x2(BR)(t_i), i.e. x has infinite p_i-height in BR. We may assume thatx2BZ[X]. ThenZ1

pi

xBR andÿ Z1

pi

x‡B

Z[X]

= BZ[X]

… † is a pi-torsion group. We infer that Z1 p_i

xBÿ Z[X]‡ Z1

pi

Li

ˆBÿ

TiZ1 pi

Li

with Ti free abelian. It follows that Z1

p_i

xBZ 1 p_i

L_i and the claim follows. p

We are now ready for the following:

THEOREM3. Let B be a torsion-free abelian group and P an infinite set of prime integers such that B is p-reduced for all p2P. Let R be a (GHS)- E-ring with prime number set P. Then the natural mapa:B!B_ZR is an absolute localization of B.

PROOF. Note thata(x)ˆx1 for allx2B. It follows from Lemma 2 that BR is an E(R)-module. By Proposition 1.2 in [6] we obtain that End_Z(BR)ˆEnd_R(BR). By Proposition 1.1 in [6] the mapais a lo- calization ofB.

6. Examples

The following is a well-known and highly instructive example.

EXAMPLE1. Even if A is p-reduced, the tensor product AA might not be p-reduced:

Letpbe prime,Jp the ring ofp-adic numbers,Zpthe ring of integers localized atp, andp2JpÿZp, a unit. Thus

pˆ lim

n!1z_n;

z_n2Z_p, in thep-adic topology withp6 jz_nandpⁿj(pÿz_n) inJ_p.

Let Aˆ h1;piJp, the pure subgroup of Jp generated by 1 and p.

ThenAAˆ h11;1p;p1;ppi is a pure subgroup of rank 4 of JpJp.

Since ppÿpz_nˆp(pÿz_n) we have pⁿj(ppÿpz_n). Simi- larly pⁿj(ppÿz_np). Hence pⁿj[(ppÿpz_n)ÿ(ppÿz_np)]ˆ (1pÿp1)z_n. Then there must exista_nandb_nin the integers such that 1ˆznan‡pⁿb_n. Hencepⁿj(1pÿp1)znanˆ(1pÿp1)(1ÿpⁿb_n) which impliespⁿj(1pÿp1) for alln<v.

SinceAis aZ_p-module this means 06ˆ1pÿp1 is a divisible ele- ment. HenceAAis not reduced.

EXAMPLE2. Even if A is strongly indecomposable, the tensor product AA might be completely decomposable:

Given distinct primes p, q, r, defineAˆe1Z1 p

e2Z1 q

‡(e1‡e2)Z1 r .

Then A is strongly indecomposable of rank 2 with End(A)ˆZ. We have

AAˆb₁Z1 p

b₂Z 1 pq

b₃Z 1 pq

b₄Z1 q

‡

‡(b1‡b3)Z 1 rp

‡(b2‡b4)Z[1

rq]‡(b1‡b2)Z 1 pr

‡

‡(b3‡b4)Z 1 qr

‡(b1‡b2‡b3‡b4)Z1 r

whereb1ˆe1e1; b2ˆe1e2; b3ˆe2e1; b4ˆe2e2. Definerbyr(b₁)ˆb₁‡b₂;andr(b_i)ˆ0 for 2i4. Then r(AA)ˆ(b1‡b2)Z1

p

‡(b1‡b2)Z 1 rp

‡

‡(b₁‡b₂)Z 1 pr

‡(b₁‡b₂)Z1 r

ˆ(b₁‡b₂)Z 1 pr

.

and thusr(AA)AA, i.e.r2End(AA) withr²ˆr.

Definesbys(b4)ˆb3‡b4;ands(bi)ˆ0 for 1i3. Then s(AA)ˆ(b₃‡b₄)Z 1

qr

‡(b₃‡b₄)Z 1 qr

‡

‡(b3‡b4)Z1 q

‡(b3‡b4)Z1 r

ˆ

ˆ(b3‡b4)Z 1 qr

AAand thus s2End(AA); s²ˆs.srˆrsˆ0.

Note that (1ÿrÿs)(b_i)ˆ

ÿb2 foriˆ1 b₂ foriˆ2 b₃ foriˆ3 ÿb3 foriˆ4 8>

>>

><

>>

>>

:

and (1ÿrÿs)(AA)ˆ ÿb₂Z1 p

‡b₂Z 1 pq

‡b₃Z 1 pq

ÿb₃Z1 q

‡

‡(ÿb2‡b3)Z 1 rp

‡(b2ÿb3)Z[1

rq]‡0‡0‡0ˆ

ˆb₂Z 1 pq

‡b₃Z 1 pq

‡(b₂ÿb₃)Z 1 pqr

ˆ:C.

Observe thatCˆ(b₂ÿb₃)Z 1 pqr

b₂Z 1 pq

, and thus

AAˆ(b1‡b2)Z 1 pr

|‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚}

im(r)

(b3‡b4)Z 1 qr

|‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚}

im(s)

(b2ÿb3)Z 1 pqr

b2Z 1 pq

|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}

im(1ÿrÿs)ˆC

is

completely decomposable.

7. Torsion-free Abelian Groups of Finite Rank with Small p-rank Again, letJ_pdenote the ring ofp-adic integers and ``'' means ``_Z''.

We viewJ_pJ_pas a rightJ_p-module. (Of course,J_pJ_pis also a left Jp-module, but we need to pick a side.)

LetDdenote the divisible part of the torsion-free groupJpJp. There is a natural (surjective) mapm:JpJp!Jpwithm(ab)ˆab. SinceJp

isp-reduced, we haveDker (m).

Let wˆ P

1ina_ib_iˆ P

1in(a_i1)b_i2ker (m) and vˆ P

1in(1a_iÿ ai1)bi. Then v2D as shown in Example 1. Note that 0ˆ1m(w)ˆ

1 P

1inaibi

ˆ P

1in(1ai)biˆw‡v. This shows that wˆ ÿv2D and thusDˆker (m).

We record this as:

REMARK2. We have JpJpˆD(1Jp)andm:JpJp!Jpis a surjective homomorphism such that Dˆker (m) is the divisible part of J_pJ_p.

LetGbe a torsion-free group andZ_pthe ring of integers localized at the primep. Then eachy2GZ_phas the formyˆa1

nfor somea2G and integernrelatively prime top. An easy argument shows thatGisp- reduced if and only ifGZ_pisp-reduced.

LetAbe a torsion-free,p-reduced group of, say, rank 2 andp-rank 1. It is easy to see that there exists ap-pure subgroupA⁰ofJpsuch that 12A⁰ andAA⁰.

NOTATION 1. Let G be a p-reduced, torsion-freeZ_p-module of finite rank n‡m and nˆrp(G)ˆdim_Z=pZ(G=pG)for some prime p. Then G

can be represented as a pure subgroup of L

1ineiJp generated by Bˆ L

1ine_iZ_p together with elements u_j ˆ P

1ine_ip_ij for 1jm and P ˆ[p_ij]2Mat_nm(J_p). We callPa representing matrix for G. Of course, the matrixP is not uniquely determined by G. Moreover, we may, and will, make the convention thatpijˆ0wheneverpij2Zp.

Fomin [8] has shown that if the entries of the matrixPare algebraically independent (overZp) andGGis notp-reduced butGisp-reduced, then rp(G)ˆ1. The following example shows that this rather strong hypothesis onP is needed:

EXAMPLE3. There exists a p-reduced torsion-freeZ_p-module G such that rank(G)ˆ4, rp(G)ˆ2and GG is not p-reduced. Moreover, G does not contain a pure subgroup A with rp(A)ˆ1and rank(A)ˆ2.

PROOF. Pick an odd prime p and a;b;g2N. such that 

pa , 

pb

 ,

p 2g JpÿZp and the field extensions Q[ 

pb

; 

pa

] and Q[ 

pb

;pg] have dimension 4 over Q with pg2=Q[ 

pb

; 

pa

]. Let uˆe1 

pa

e2 

pbg and vˆe₁ 

pab

e₂pg. Let Gˆe₁Z_pe₂Z_p;u;v

e₁J_pe₂J_p. Let wˆ e₁e₁a(bÿ1)‡e₂e₂g(1ÿb)‡uuÿvv2GG L

1i;j2J_p(e_ie_j)J_p. We claim that

wis ap-divisible element ofG:

Note thatwˆ

e1e1a(bÿ1)‡ 

pa

e1e1 

pa

ÿ 

pab

e1e1 

p ab

‡

‡ pa

e₁e₂ 

pbg

ÿ 

pab

e₁e₂pg

‡

‡ 

pbg

e₂e₁ 

pa

ÿpge₂e₁ 

p ab

‡

‡

e₂e₂g(1ÿb)‡ 

pbg

e₂e₂ 

pbg

ÿpge₂pge₂ :

Each term in a square bracket is in the divisible part ofJpe_ie_jJpand thusw6ˆ0 is a divisible element ofGG.

Now let 06ˆgˆe₁a₁‡e₂a₂‡ub‡vc2G. We claim thatgp2=Gfor all p2J_pÿZ_p. This allows us to infer thatGhas no pure subgroupAwith rank(A)ˆ2 andp-rankrp(A)ˆ1.

Letg⁰ˆe1a⁰₁‡e2a⁰₂‡ub⁰‡vc⁰2Gsuch thatgpˆg⁰.

W.l.o.g. we may assume that all coefficients are integers. Using matrix notation, we get

a₁‡b 

pa

‡c 

pab a2‡b 

pbg

‡cpg

" #

pˆ a⁰₁‡b⁰ 

pa

‡c⁰ 

pab a⁰₂‡b⁰ 

pbg

‡c⁰pg

" #

.

Case 1:a1‡b 

pa

‡c 

pab

6ˆ06ˆa2‡b 

pbg

‡cpg. In this case we havep2Q[ 

pa

; 

pb

]\Q[ 

pb

;pg]ˆQ[ 

pb

] and w.l.o.g.

we may assume thatpˆ 

pb

and we get a₁‡b 

pa

‡c 

pab a2‡b 

pbg

‡cpg

" #

b p ˆ

b

p a1‡b 

pab

‡cb 

pa

b

p a₂‡c 

pbg

‡bbpg

" #

ˆ a⁰₁‡b⁰ 

pa

‡c⁰ 

pab a⁰₂‡b⁰ 

pbg

‡c⁰pg

" #

and we infer the equations:

a₁ˆ0ˆa⁰₁;bˆc⁰;cbˆb⁰as well as

a₂ˆ0ˆa⁰₂;cˆb⁰;bbˆc⁰ and thus bbˆb andcbˆc. We now have thatbˆ0ˆcandgˆ0, a contradiction.

Case 2:a₁‡b 

pa

‡c 

pab

ˆ0 buta₂‡b 

pbg

‡cp 6ˆg 0 (or vice versa).

In this case we have that a⁰₁ˆb⁰ˆc⁰ˆ0ˆa₁ ˆbˆc and thus a₂pˆa⁰₂, a contradiction top2=Z_p. p A possible choice for p;a;b;g satisfying our hypotheses would be pˆ19, aˆ7, bˆ11 and gˆ17, because these are distinct primes and 8²7 mod 19, 7²11 mod 19 and 6²17 mod 19. Therefore, by Hensel's Lemma, we have that 

p7

; 

p11

; 

p17

2J₁₉.

Next we will introduce a quasi-isomorphism invariant for our groupG.

Let Gˆe_iZp;u_j;1in;1jk

L

1ine_iJp be a p-reduced group of rankn‡kandrp(G)ˆnwhereujˆ P

1ineipijwithpij2Jp. Let G⁰ˆDe⁰_iZp;u⁰_j;1in⁰;1jk⁰E

L

1ine⁰_iJpbe another such group withu⁰_jˆ P

1in⁰e⁰_ip⁰_ijfor 1jk⁰.

Assume that for allb;1bn⁰, the setsPbˆ f1;p⁰_bj :1jk⁰gare linearly independent over the field extensionKˆQÿp_ij;1in;1jk

. ThenGG⁰isp-reduced:

We haveGG⁰ˆL

i;jeie⁰_jZp‡P

i;jeiu⁰_jZp‡P

i;juie⁰_jZp‡P

i;juiu⁰_jZp. Letyˆ P

1i;jneie⁰_jaij‡P

i;jeiu⁰_jbij‡P

i;juie⁰_jcij‡P

i;juiu⁰_jdij2GG⁰.

Note that theeae⁰_bentry ofyis y_abˆeae⁰_ba_ab‡P

j eae⁰_bp⁰_bjb_aj‡P

i p_aieae⁰_bc_ib‡P

i;j p_ai(eae⁰_b)p⁰_bjd_ij: Ifyis a divisible element, then, by Remark 2, we have

0ˆm(y_ab)ˆa_ab‡P

j p⁰_bjb_aj‡P

i p_aic_ib‡P

i;jp_aip⁰_bjd_ijˆ

ˆ

aab‡P

i paicib

‡P

j p⁰_bj baj‡P

i paidij

for alla;b.

By our hypothesis that f1;p⁰_bj:1jk⁰g is linearly independent over the field extension KˆQÿp_ij;1in;1jk

, we infer that aab‡P

i paicibˆ0 and it follows thatP

i uicib2 L

1jnejZp. This is a con- tradiction ton‡kˆrank(G) unless alla_ab ˆ0ˆc_ib.

We infer that P

j p⁰_bj(baj‡P

i paidij)ˆ0 >for all a;b and thus baj‡P

i paidijˆ0 for alla;j. Again, it follows thatP

i uidij2 L

1gnegZpand thusd_ijˆ0 for alli;j. Now we infer thatb_ajˆ0 for alla;jand we have that yˆ0.

PROPOSITION 8. Let Gˆe_iZp;u_j;1in;1jk

L

1ine_iJp

be a p-reduced group of rank n‡k and rp(G)ˆn where u_jˆ P

1ine_ip_ij with pij2Jp. Let G⁰ˆ

e⁰_iZp;u⁰_j;1in;1jk

L

1ineiJp be another such group with u⁰_jˆ P

1ineip⁰_ij. Let K (K⁰)be the field extension ofQgenerated by the elementsp_ij(p⁰_ij).

If G is quasi-isomorphic to G⁰, then KˆK⁰. PROOF. Note that GˆD L

1ine_iZ_p

L

1iku_iZ_pE

L

1ine_iJ_p whereu_iˆ P

1jne_jp_jifor somep_ji2J_p. PutP ˆ[p_ji]2Mat_nk(J_p).

Let W:G!G⁰ be a quasi-isomorphism. Then there exist matrices Aˆ[aij]2Matnn(Q);Bˆ[bij]2Matkn(Q) such that

W(e_i)ˆP

j e⁰_ja_ji‡P

j u⁰_jb_ji. Now

W(ui)ˆP

j W(ej)pjiˆP

j

P

a e⁰_aaaj‡P

b u⁰_bbbj

pjiˆ

ˆP

j

P

a e⁰_aa_aj‡P

b

P

a

ÿe⁰_ap⁰_ab b_bj

p_jiˆ

ˆP

a e⁰_a P

j

aajpji‡P

b p⁰_abbbjpji

ˆ

ˆP

a e⁰_a…[AP]_ai‡[P⁰BP]_ai†.

SinceW(u_i)2G⁰for all 1ik, we have that W(u_i)ˆP

j e⁰_jx_ji‡P

j u⁰_jy_jiˆP

a e⁰_ax_ai‡P

j

ÿ P

a e⁰_ap⁰_aj y_jiˆ

ˆP

a e⁰_a

x_ai‡P

j p⁰_ajy_ji

. LetXˆ[x_ai]2Mat_nk(Q) and Yˆ[yai]2Matkk(Q). We now have a matrix equation

AP‡P⁰BPˆX‡P⁰Y ofnkmatrices, which is equivalent to P⁰(BPÿY)ˆXÿAP whereBPÿY is a squarekkmatrix.

Note that W(ei)ˆP

j e⁰_jaji‡P

j u⁰_jbjiˆP

a e⁰_aaai‡P

j

P

a e⁰_ap⁰_ajbjiˆP

a e⁰_a(A‡P⁰B)_ai. SinceWis injective, the square matrix A X

B Y

2Mat_(n‡k)(n‡k)(Q) is invertible.

Assume that there is someT2Mat_kk(J_p) such that (BPÿY)Tˆ0:

Then 0ˆP⁰(BPÿY)Tˆ(XÿAP)T.

This implies that A X B Y

PTÿT

ˆ APTÿXT BPTÿYT

ˆ 0

0 . Since A XB Y

is invertible, we infer thatTˆ0. It follows thatBPÿY is in- vertible.

Let KˆQÿp_ij;1in;1jk

and K⁰ˆQÿ

p⁰_ij;1in;1jk . Then BPÿY2Matmm(K) and so is (BPÿY)^ÿ1, which implies P⁰2 Mat_kk(K) and thusK⁰K. By symmetry it follows thatKˆK⁰. p LetAbe an E-ring. LetW2Hom(AA;A). The groupAAcontains subgroups of the form Ab and aA. There exists a surjective map i_b:A!Ab with i_b(x)ˆxb for all x2A. Then c_bˆWi_b:A!A

and since A is an E-ring, there is some sb2A such that c_b(x)ˆ W(xb)ˆxsb for all x2A. By a similar argument, there are elements pa2Asuch thatW(ay)ˆypa for ally;a2A.

Note that W(ab)ˆbp_aˆs_ba. It follows that s₁ˆp₁. Foraˆ1 we gets_bˆbp₁and thusW(ab)ˆabp₁for alla;b2A.

We conclude thatWˆp1m where m2Hom(AA;A) is the map with m(ab)ˆabfor alla;b2A. We have shown:

PROPOSITION9. If A is an E-ring, then the mapm:AA!A with m(ab)ˆab for all a;b2A is a localization.

8. Zero Product Determined Algebras

The following definition can be found in [4] and elsewhere:

DEFINITION4. Let A be an algebra over the commutative ring C. Then A is calledzero product determinedif for any C-module X and every C- bilinear maph j i:AA!X the following holds:

If for all a;b2A, abˆ0 implies ah jbi ˆ0, then there exists some T2Hom_C(A²;X)with xh jyi ˆT(xy)for all x;y2A. As usual A²denotes the C-submodule of A generated by the set of products xy for x;y2A.

REMARK3. Let C be a commutative ring and X some C-module. We will always assume that X is a C-bimodule with sxˆxs for all x2X and all s2C. Let X;Y;W be C-modules. Then each C-linear map

j

h i:XY!W is automatically a middle linear map, i.e. xsh jyi ˆ xjsy

h i for all x2X;y2Y and s2C. This means that h j i factors through X_CY. We will write ``'' instead of ``_C''.

PROPOSITION 10. Let A be a C-algebra. Consider the maps AA! AA!^m A²ˆspanCfxy:x;y2Agwherem(xy)ˆxy.

The following are equivalent:

(a) Let X be some C-module andu2HomC(AA;X). Ifker (m ) ker (u ), then there exists some T2Hom_C(A²;X)making the following diagram commutative: AA ! AA !^m A²

#u #T

X

:Note that if 12A, then AˆA².

(b) A is zero product determined.

(c) Let X be some C-module andu2HomC(AA;X). Ifker (m ) ker (u ), thenker (m)ker (u).

PROOF. Assume that (a) holds and leth j i:AA!X be aC-bi- linear map such thathajbi ˆ0 wheneverabˆ0. Then there exists a un- iqueu2HomC(AA;X) such thathajbi ˆu(ab) for alla;b2A. Thus u(ab)ˆ0 wheneverabˆ0, i.e. ker (m )ker (u ). By clause (a), there is some T2Hom_C(A²;X) such that uˆTm. It follows that

ajb

h i ˆ(Tm)(ab)ˆT(ab) for alla;b2Aand (b) holds.

For the converse, assume that (b) holds. With the notations in (a), define h j i:AA!Xbyhajbi ˆu(ab) for alla;b2A. Since ker (m ) ker (u ) we have thatabˆ0 implieshajbi ˆ0 for alla;b2A. By clause (b), there exists some T2Hom_C(A²;X) such that hajbi ˆT(ab) for all a;b2A. Thereforeu(ab)ˆhajbi ˆT(ab)ˆT(m(ab)) for alla;b2A.

This shows thatuˆTmand (a) follows.

That (a) implies (c), is a trivial consequence of the commutative diagram in (a). For the converse, if ker (m)ker (u), the mapTmay be defined by Tÿ P

i a_ib_i

ˆuÿ P

i a_ib_i

, sinceP

i a_ib_iˆ0 meansmÿ P

i a_ib_i

ˆ0. p

Note that the mapTin clause (a) is unique since the mapmis surjective.

DEFINITION5. Let C be a commutative ring and A a C-algebra and X a C-module. Then Hom⁰_C(A;X)ˆ fu2Hom_C(AA;X): If x;y2A with xyˆ0, thenu(xy)ˆ0g.

Then A is zero product determined if and only if for any u2Hom⁰_C(A;X) there exists a (unique) T2Hom_C(A²;X) such that uˆTm.

Note that for any T2Hom_C(A²;X) we have that T(m(xy))ˆ T(xy)ˆT(0)ˆ0 whenever xyˆ0, i.e. Tm2Hom⁰_C(A;X). This shows that

Hom_C(A²;X)!^mHom⁰_C(A;X)!0 is exact. We have shown:

PROPOSITION11. The C-algebra A is zero product determined if and only if Hom_C(A²;X)Hom⁰_C(A;X)via the natural map m.

Assume that 12A has no zero divisors and is zero product de- termined. Then Hom⁰_C(A;X)ˆHom_C(AA;X)Hom_C(A;X) via the natural map m.

PROPOSITION12. Let C be an integral domain and A a torsion-free C-algebra with 12A and without zero divisors. Then A is a zero product determined algebra if and only if A is a subring of the field of fractions of C.

PROOF. Note that ker (m )ˆ f(a;0):a2Ag [ f(0;a):a2Ag ker (u ) for all u2Hom_C(AA;X). Consider the identity map id:AA!AA. If Ais zero-product determined, then there exists a map T:A!AA such that idˆTm and thus m is an isomorphism.

This shows that theC-rank ofAis 1 and thusAis a subring of the field of

fractions ofC. The converse is obvious. p

COROLLARY 6. The ring J_p of p-adic integers is not zero product determined.

This motivates a weaker condition:

DEFINITION6. Let C be a commutative ring and A a C-algebra. We call A zero product self-determined, if A has the property (a) of Proposition11 with ``X'' replaced by ``A²''.

THEOREM4. Let A be an integral domain. The following are equiva- lent:

(a)The ring A is a zero product self-determinedZ-algebra.

(b)The mapm:AA!A is a (surjective) localization of AA.

(c)The ring A is an E-ring.

PROOF. Because of Proposition 9 and 10, we only have to show that (b) implies (c). To this end, pick anyb2End(A) and consideridAb:AA! AA. Then there exists someT:A!Awithm(id_Ab)ˆTmand thus ab(b)ˆT(ab) for alla;b2A. Letaˆ1. Thenb(b)ˆT(b) follows for allb2B and thusab(b)ˆb(ab) for all a;b2A. Now, for bˆ1, we get b(a)ˆab(1) for alla2A. Sinceb2End(A) was arbitrary, it follows thatAis an E-ring.

Note that the hypothesis that A has no zero divisors was not used in the

previous argument. p

We may use the previous argument and Proposition 9 to obtain the following characterization of E-rings:

COROLLARY 7. Let A be some ring (viewed as a Z-algebra). The following are equivalent:

(a)The natural mapm:AA!A is a localization.

(b)The ring A is an E-ring.

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Manoscritto pervenuto in redazione il 29 Giugno 2013.