Endoprimal abelian groups of torsion-free rank 1

Endoprimal Abelian Groups of Torsion-Free Rank 1.

K. KAARLI(*) - L. MA´_RKI(**)

ABSTRACT - This paper completely solves the endoprimality problem of (mixed) abelian groups of torsion-free rank 1.

1. Introduction.

In an abelian group A, the functions of the form f(x₁,R,x_n)4k₁x₁1R1k_nx_n

with integersk1,R,kn are called theterm functions. Clearly, term func- tions permute with all endomorphisms ofA, i.e., iff(x1,R,x_n) is a term function and f is an endomorphism of A then

f(f(a₁,R,a_n) )4f(f(a₁),R,f(a_n) )

for anya₁,R,a_nA. More generally, we call a functionfof finite arity in an abelian groupA anendofunction if it permutes with all endomor- phisms of A. An abelian group is called endoprimal if all its endofunc- tions are term functions, in other words, if the term functions are the

(*) Indirizzo dell’A.: Institute of Pure Mathematics, University of Tartu, 50090 Tartu, Estonia. E-mail: kaarliHmath.ut.ee

(**) Indirizzo dell’A.: A. Rényi Institute of Mathematics, Hungarian Acade- my of Sciences, H-1364 Budapest, Pf. 127, Hungary. E-mail: markiHrenyi.hu

The first author thanks the Estonian Science Foundation grant no. 5368 for support.

The second author’s research was partially supported by Hungarian National Foundation for Scientific Research grants T34530 and T43034.

Both authors thank the Estonian and the Hungarian Academies of Sciences for making possible mutual visits through their exchange agreement.

only functions in the group which permute with all endomorphisms.

The notion of endoprimality comes from universal algebra, where it has emerged in two ways: as a generalisation of an important property (called primality) of the two-element Boolean algebra and in the course of investigations into a general duality theory. The investigation of abe- lian groups with respect to endoprimality was started by Davey and Pit- kethly [1] who, beside results for other kinds of algebraic structures, de- scribed those bounded abelian groups which are endoprimal.

As is well known, classification results for abelian groups, like for most algebraic structures, can only be obtained under severe restric- tions. Obviously, endoprimality is a strong condition also for abelian groups and thus it seems promising to find characterisations for these groups. In [5] the present authors pointed at the connections of endopri- mality with (direct) decomposition properties of abelian groups. Among others, they proved that a torsion group is endoprimal if and only if it is of finite exponent m containing Z_m5Z_m as a subgroup, and that any group of the formB5ZwithBunbounded is endoprimal, and also cha- racterised endoprimal rank-2 torsion-free groups by means of a decom- position property. Investigations of endoprimality in torsion-free groups have been continued in [4] and [6]. In [4], endoprimal torsion-free sepa- rable groups are characterised, but also arbitrarily large indecomposa- ble endoprimal groups are presented. In [6], endoprimal rank-3 torsion- free groups are described, and even among these there are indecompo- sable ones.

In the present paper we take up a further line from [5]. There the en- doprimality problem was solved for the mixed groups of torsion-free rank 1 with splitting torsion part. To formulate that result and for fur- ther use, let us fix some notations.

In what follows, all groups are abelian and T4T(A) always stands for the torsion part of a groupA. Moreover, for any primep,Tp4Tp(A) is thep-component ofTandT_p* is the sum allT_q,qcp. The fact thatBis a direct summand ofAis denoted byB±A. ByJ_pwe denote the ring of p-adic integers.

So we have:

THEOREM 1.1 ([5], Theorem 5.9). Let A be a group and P be the set of those primes p for which A/T is p-divisible. Assume that A has tor- sion-free rank1 and T±A. Then A is endoprimal if and only if T is unbounded and, for every pP, the subgroup T_p is not reduced.

Now we are going to settle the mixed torsion-free rank 1 case com- pletely and prove the following theorem as the main result of the paper.

Notice that we also get endoprimal groups in which none of the primary parts split off; in other words, they are as close to being indecomposable as a mixed group of torsion-free rank 1 can be.

THEOREM 1.2. Let A be a group and P be the set of those primes p for which A/T is p-divisible. Assume A has torsion-free rank 1. Then A is endoprimal if and only if T is unbounded and, for every pP,T_p is either not reduced or it is not a direct summand of A.

2. Unary endofunctions.

The proof of the first three lemmas is straightforward.

LEMMA 2.1. Every endofunction of a group A preserves the kernels and images of all endomorphisms of A. r

LEMMA 2.2 ([5], Proposition 2.5). Let A4A15A2and f be an n-ary endofunction of A.Then f_i,the restriction of f to A_i,is an endofunction of A_i, i41 , 2 , and

f(x11y1,R,xn1yn)4f1(x1,R,xn)1f2(y1,R,yn) holds for arbitrary x_iA₁, y_iA₂, i41 ,R,n. r

LEMMA 2.3. Let A be a group and f a unary endofunction in A.If b and c are endomorphic images of the same element aA then f(b1c)4f(b)1f(c). r

LEMMA 2.4. If f is any unary endofunction on a group A then fN^Tis an endomorphism of T.Moreover, for every p-component T_pof T,there exists jJ_p such that f(x)4jx for every xT_p.

PROOF. We first consider the action offon thep-componentT_p. We start with the case when T_p is not reduced, that is, it has a quasicyclic subgroupD. Obviously,Dis a direct summand ofA; letA4B5D. Then by 2.2 there exist endofunctionsfBandfDof the groupsBandD, respec- tively, such that for everyxB,yDwe havef(x1y)4f_B(x)1f_D(y). It is an easy exercise to show that the unary endofunctions ofDare preci- sely its endomorphisms, that is, multiplications byp-adic integers. Letj

be the p-adic integer such that f(x)4jx for every xD. Now, if D₁ is any other quasicyclic subgroup of T_p then there exists fEndA that maps D isomorphically onto D1. This yields that f(x)4jx holds for all xD1, but then also for any xE where E is the divisible part of T_p. Let A4B₁5E,B₂4B₁OT_p, thusB₂is a reduced p-group. If B₂is bounded then it has a cyclic direct summand C of A of highest order.

Now there existsfEndAthat embedsCintoD, implyingf(x)4jxfor everyxC. Also, every cyclic subgroup ofB2is an endomorphic image of C, hencef(x)4jxholds for everyxB2. Now, ifxB2andyEare ar- bitrary then the projections ofAtoB₁andE takex1y tox andy, re- spectively, thus by Lemma 2.3 we obtain

f(x1y)4f(x)1f(y)4jx1jy4j(x1y) .

Now assume that B2 is unbounded. A reduced unbounded p-group has cyclic direct summands of arbitrarily high orders (see e.g. [3], Exer- cise 27.1). These direct summands, being pure subgroups ofT_p, are pure subgroups of A. Then, as bounded pure subgroups, they all are direct summands of A. Thus, A has cyclic direct summands C_i of arbitrarily high orders pⁿⁱ. Since any of these subgroups can be embedded by means of an endomorphism into D, it follows that f(x)4jx holds for everyxin anyC_i. Since any cyclic subgroup ofT_pis an endomorphic ima- ge of some C_i, we have f(x)4jx for all xT_p.

The case whenT_pis reduced can be handled similarly. IfT_pis bound- ed then it has a cyclic direct summandC of highest order andCis a di- rect summand ofAas well. This yields that the restriction offtoCis an endofunction ofC, thus there is an integernsuch thatf(x)4nxfor any xC. Also, for every cyclic subgroupC₁ofT_pthere existsfEndAthat maps C onto C₁. This yields that f(x)4nx holds for any xT_p.

To finish withT_pwe need to handle the case whenT_p is reduced but not bounded. As above, we find cyclic direct summandsCiofAcontained inT_pof strictly increasing orders and on each of them the functionfcan be computed byf(x)4n_ixwheren_iis an integer. Because for every i,j with iEj there is an endomorphism of A which embeds C_i into C_j, all thesen_i’s can be replaced by a single p-adic integer j. Thus, f(x)4jx holds for everyi and every xCi. Since any cyclic subgroup ofTp is an endomorphic image of some Ci, the same formula holds for any xTp. By this we have, in particular, that f acts as an endomorphism in every T_p.

Next we prove the equality f(x1y)4f(x)1f(y) for the case when

x,yT belong to different p-components. By the foregoing we have f(x)4mx and f(y)4ny for some integers m,n. We first find k,lZ such that

kx4x, ky404lx, ly4y.

The orders of f(x) and f(y) divide the orders of x and y, respectively, hence

k(f(x1y) )4f(k(x1y) )4f(x)4mx4k(mx1ny)4k(f(x)1f(y) ) and similarlyl(f(x1y) )4l(f(x)1f(y) ). Since obviously kandl can be chosen coprime, we have f(x1y)4f(x)1f(y).

Now, using an easy induction argument one can prove the formula f(x11R1xn)4f(x1)1R1f(xn)

(1)

for arbitraryx₁,R,x_nbelonging to pairwise different p-components ofT.

To conclude the proof, we prove the equality f(x1y)4f(x)1f(y) for arbitrary elements x,yT. Assume that x,yT_p11R1T_pn and x4x₁1R1x_n, y4y₁1R1y_n where x_i,y_iT_pi, i41 ,R,n. Then, using the formula (1) and the fact that finduces endomorphisms on all p-components, we have

f(x1y)4f( (x11R1x_n)1(y11R1y_n) ) 4f( (x11y1)1R1(x_n1y_n) ) 4f(x₁1y₁)1Rf(x_n1y_n)

4(f(x₁)1f(y₁) )1R1(f(x_n)1f(y_n) ) 4f(x₁1R1x_n)1f(y₁1R1y_n) 4f(x)1f(y) . r

LEMMA 2.5. Let A be an abelian group with torsion part T,and as- sume that rank (A/T)41 . Then every unary endofunction of A is an endomorphism.

PROOF. Consider an arbitrary unary endofunctionfofA. By Lemma 2.4,fpreservesTand the restriction of ftoTis an endomorphism ofT. Next we show that, for anyaA0TandtT,f(a1t)4f(a)1f(t). First observe that the general case of this formula can be derived by an induc-

tion argument from the special one withtT_p. Thus, assumetT_p. Mo- reover, for the beginning assume thattis contained in a direct summand CofA. LetA4B5C,a4b1t1,bB,t1C. Then it follows from Lem- ma 2.2 thatf(a)4f(b)1f(t1) andf(a1t)4f(b)1f(t11t). Hence, using Lemma 2.4 we have

f(a1t)4f(b)1f(t₁1t)4f(b)1f(t₁)1f(t)4f(a)1f(t) . Consequently, the formula f(a1t)4f(a)1f(t) holds whenever T_p is a direct summand of A.

It remains to consider the case whenTp±O A. In this caseTpcontains cyclic direct summands of A of arbitrarily high orders. We take one of order at least the order oft. Let it beC4acb. LetA4A₁5C,a4b1u wherebB,uC. Now both a andt are endomorphic images of b1c, and Lemma 2.3 gives f(a1t)4f(a)1f(t).

Finally, if a₁,a₂A0T then there exist aA0T, m₁,m₂Z, t₁, t2T such that a14m1a1t1, a24m2a1t2. Hence

f(a₁1a₂)4f( (m₁1m₂)a1(t₁1t₂) ) 4f( (m₁1m₂)a)1f(t₁1t₂) 4(m11m2)f(a)1f(t1)1f(t2) 4(m1f(a)1f(t1) )1(m2f(a)1f(t2) ) 4(f(m₁a)1f(t₁)1(f(m₂a)1f(t₂) ) 4f(m₁a1t₁)1f(m₂a1t₂)

4f(a₁)1f(a₂) . r

Call a group 1-endoprimal if those of its unary functions which per- mute with all endomorphisms are exactly the unary term functions, that is, the functions of the form kx with some integer k. Now we have:

COROLLARY 2.1. An abelian group of torsion-free rank 1 is1-endo- primal if and only if it is not p-divisible for any prime p and its endo- morphism ring has trivial centre. r

3. Proof of the Main Theorem.

The following statement is a direct consequence of [7], Proposition 2.2 or even of [2], Theorem 1.2.

LEMMA 3.1. Let A be a group of torsion-free rank 1. If T4T_p and A/T is not p-divisible then T±A. r

For easy reference, we state without proof the following simple fact.

LEMMA 3.2. For any group A and prime number p: T/T_p* ±A/T_p* ¨ T_p±A. r Lemmas 3.1 and 3.2 directly imply

LEMMA 3.3. Let A be a group of torsion-free rank 1. If A/T is not p-divisible then T_p ±A. r

The most unexpected conclusion from Theorem 1.2 is that a group of torsion-free rank 1 is endoprimal provided none of its nonzerop-compo- nents splits off and it has a non-zerop-component for every primepsuch that the torsion-free factor of the group isp-divisible. This fact is based on the next lemma.

LEMMA 3.4. Let A be a group of torsion-free rank 1 and let Tp be reduced. Assume that there exist jJ_p and fEndA such that 0cf(A)’T_p and f(x)4jx for every xT_p. Then T_p±A.

PROOF. Lemma 3.2 reduces the proof of the present lemma to the ca- seT4Tp. If A/T is not p-divisible then T4Tp ±A follows by Lemma 3.1. Assume that A/T is p-divisible and consider first the case j40 . Clearly thenT’Kerfand we have a homomorphismf:A/TKT. Hence f(A/T)4f(A) is a divisible subgroup ofT. SinceAis reduced,f(A)40 , a contradiction.

Ifjc0 thenjcan be written in the formj4p^khwherehis an inver- tible p-adic integer and kF0 . The function xOh²¹x is a well-defined endomorphism of T_p, hence we have c4h²¹fEndA and obviously c(x)4p^kx for every xT. Moreover, f(A)’T implies c(x)4p^kx iff xT. Now, ifxEndAis defined byx(x)4c(x)2p^kxthen Kerx4T, hencex(A)`A/T. Consequently, x(A) is a torsion-free subgroup of A, that is, x(A)OT40 .

We show thatx(A)1T4A, which proves T±A. LetA4A/Tandx be the endomorphism ofAinduced byxmoduloT. Thenx(A)4p^kA4A becauseAisp-divisible. Hence,xis surjective, which is equivalent to the

statement thatx(A) intersects all cosets ofTin A. The latter is equiva- lent to the equality x(A)1T4A. r

We need one more lemma which helps to carry out the induction in the proof of the main theorem. Note that actually the same idea was used several times in [5].

LEMMA 3.5. Let A4B5C. Assume that every cC is contained in some subgroup f(B) where fEndA. Then, if the restriction of every unary endofunction of A to C is a term function, the same holds for endofunctions of A of arbitrary arity.

PROOF. Suppose that the restrictions of all endofunctions ofAof ari- ty less thannare term functions. Letfbe ann-ary endofunction ofAand c₁,R,c_nC. We takefEndAsuch thatf(C)40 andc₁4f(b) where bB. Also, letcEndA be the projection of A onto C composed with the natural embedding of C into A. Then

f(c1,R,cn)4f( (f1c)(b), (f1c)(c2),R, (f1c)(cn) ) 4(f1c) f(b,c₂,R,c_n)

4f(f(b,c₂,R,c_n) )1c(f(b,c₂,R,c_n)

4f(f(b),f(c2),R,f(cn) )1f(c(b),c(c2),R,c(cn) ) 4f(c₁, 0 ,R, 0 )1f( 0 , c₂,R,c_n) .

Now our claim follows directly from the induction hypothe- sis. r

PROOF OF THEOREM 1.2. Neccessity. By [5], Corollary 5.2 the tor- sion part must be bounded. IfT_p40 forpP, then xO¹

pxis an endo- function which is not a term function, henceAis not endoprimal. Finally, assume that there existspP such that T_p is a nonzero, reduced direct summand of A and let A4B5T_p. Clearly Hom (T_p,B)40 . Let f Hom (B,Tp). Obviously, Tp* is the torsion part of B, Tp*’Kerf, and A/T`B/T_p* . Hence finduces a homomorphismc:A/TKT_p. SinceA/T isp-divisible, c(A/T) is a divisible subgroup of T_p. SinceT_p is reduced, we havec40 but then alsof40 . Thus, Hom (B,T_p)40 , and using [5], Corollary 2.6, we conclude that A is not endoprimal.

PROOF OFTHEOREM 1.2. Sufficiency.We first prove that Ais 1-en- doprimal. Letfbe a unary endofunction ofA. By Lemma 2.5fis an endo- morphism ofA. Hence fpreserves Tand induces an endomorphismf of A/T defined by f(a1T)4f(a)1T. Thus there is a rational number a such thatf(a)1T4a(a1T) for anyaA. Consider first the case when aZ. Define a new functionf₁(x)4f(x)2ax. Clearly,f₁is both an en- domorphism and an endofunction of A, and f₁(A)’T. We are going to show that f140 , which implies that f is a term function.

Given a prime numberp, letppbe the natural projection ofTontoTp

composed with the embedding ofT_pintoT. Thenp_pis an idempotent en- domorphism of T with range T_p and also an endofunction of T. Since f₁(A)’T and f₁N^TEndT, the composition f_p4ppf₁is an endofunction of A. We shall show, by checking three cases separately, that fp40 for any prime p. This will prove that f140 .

Case 1:pP. By Lemma 3.3 we haveA4B5T_p for a suitable sub- group B of A. Since f_p(A)’T_p, Lemma 2.2 implies f_pN^B40 . Obviously B/T(B)`A/T, therefore B/T(B) is not p-divisible. Hence, given any t T_p, there existsfEndAsuch thatf(b)4t, for a suitable elementb B. Since fp is an endofunction ofA, we have 04f(fp(b) )4fp(f(b) )4 4f_p(t). Thus, f_pNTp40 . Now Lemma 2.2 gives f_p40 .

Case 2:pP,T_p±AandT_pis not reduced. NowT_phas a nonzero di- visible partDand we have direct decompositions A4B5T_p,T_p4T_p85 5D. As in Case 1, we getf_pN^B40 . Due topPthe groupB/T(B) isp-divi- sible, hence, given anytD, there exist fEndA and bB such that f(b)4t. Now again as in Case 1 we concludefp(t)40 , hencefpN^D40 . It remains to show thatf_pNTp840 . First, any cyclic direct summandCof T_p8 can be embedded by an endomorphism of A into D. Since f_p permutes with that endomorphism, we get f_pN^C40 . Now there are two possibili- ties: 1)T_p8 is bounded; 2) T_p8is not bounded. In the first case we take a cyclic direct summandCofTp8of the highest order. ThenCcan be map- ped by an endomorphism of A onto any cyclic subgroup of T_p8. Hence f_pN^C40 impliesf_pN^Tp840 . In the second case we know thatT_p8has cyclic direct summands C_i, i41 , 2 ,R, of arbitrarily high orders and obviou- sly the collection of subgroupsf(C_i),fEndA,i41 , 2 ,R, includes all cyclic subgroups of Tp8. As above, we conclude that fpN^Tp840 .

Case 3:pPandTp±O A. LetDbe the divisible part ofTp,A4B5D and T_p8 4T_pOB. Then T_p8±O B, hence T_p8 is unbounded and obviously T_p8 4T_p(B) is reduced. Assume that f_pc0 . Then by Lemma 2.2 the re- striction off_p either toBorDmust be nonzero. We show thatf_pN^Bc0 .

Indeed, by Lemma 2.4 there existsjJ_p such that f_p(x)4jx for every xT_p. SinceT_p8 is unbounded,f_pN^Tp840 would implyj40 but then also fpN^D40 . Now Lemma 3.4 applies withBandfpN^Bin the roles ofAandf, respectively, to claim that T_p8±B, a contradiction.

We have finished the proof of the claim thatfis a term function ofAif aZ. In general, a can be written as an irreducible fraction

a4 m p1k₁Rpsk_s

where m,k₁,R,k_sZ, k₁,R,k_sF1 , and the p_i are prime numbers.

SincexOaxis an endomorphism ofA/T, all thep_imust belong toP. Ifp_i is a prime factor of the denominator ofa, thenTp_ic0 by the assumptions of the theorem. Let tTp_i be an element of order pi, then

p₁^k1Rp_s^ksf(t)4f(p₁^k1Rp_s^kst)4f( 0 )40 but, on the other hand,

p₁^k1Rp_s^ksf(t)4mtc0

formis prime to p_i. HenceaZcannot take place, and this completes the proof of the claim that A is 1-endoprimal.

We continue by induction on the arity of the endofunctionf. We assu- me that all endofunctions ofAwhose arity is less thannare term func- tions of A. Let f be an n-ary endofunction of A, nF2 .

We start with showing thatfN^Tpis a term function ofTpfor every pri- mep. Again we have to handle several cases separately. Obviously, we may assume that T_pc0 .

Case 1.pP. Then Lemma 3.3 implies thatT_psplits off, say,A4B5 5T_p. Since B/T_p*`A/T is not p-divisible, for every cT_p there exist bB and fEndA such that f(b)4c. Hence our claim follows by Lemma 3.5.

Case 2.pP,T_p±O A. NowT_pcontains cyclic direct summands ofAof arbitrarily high orders. This implies that, for any positive integerk, the groupAhas an endomorphic imageS_kisomorphic toZpn^k. Being an endo- function, f preserves every endomorphic image of A, moreover, the re- striction of f to that image is an endofunction of the latter. Since all groupsZ_p^nkare endoprimal, it follows that the restrictions offto all sub- groups S_k are term functions. Moreover, since for every k,l with kEl there exists an endomorphism ofAthat embedsS_kintoS_l, there existp-

adic integers j1,R,jn such that

f(x₁,R,x_n)4j1x₁1R1jnx_n (2)

for everykand allx1,R,x_nS_k,k41 , 2 ,R. Finally, ifx1,R,x_nT_p are arbitrary then there are an integerk and an endomorphism f of A such that x₁,R,x_nf(S_k). From this we conclude that the formula (2) holds for all x₁,R,x_nT_p. Since g₁(x)4f(x, 0 ,R, 0 ) is a term func- tion by the induction hypothesis andg1(x)4j1xforxTp, we conclude that j1Z. Similarly, all other coefficients ji are integers.

Case 3.pP,T_p ±A. Now it follows from our assumptions thatT_p cannot be reduced. LetDc0 be the divisible part ofT_p. Hence we have direct decompositionsA4B5T_p andT_p4C5D. We first consider the subgroup B5D. Since B/T_p*`A/T_p is p-divisible, the assumption of Lemma 3.5 is satisfied, that is, for everydDthere existbBand f EndAsuch thatd4f(b). Therefore we can conclude that the restriction of f to D is a term function; let

f(x₁,R,x_n)4a1x₁1R1anx_n (3)

for all x₁,R,x_nD and for fixed a1,R,anZ.

Now, if x₁,R,x_nT_p and f is any endomorphism of A such that f(Tp)’D, easy calculations show that

f(x₁,R,xn)2a1x12R2anxnKerf.

Thus the equality (3) will be verified for all x₁,R,x_nT_p if we prove that the intersection of the kernels of all such endomorphisms and the subgroupTp is zero. But this follows easily from the injectivity ofD. In- deed, every cyclic subgroup ofTpcan be embedded intoD. SinceDis in- jective, this embedding can be extended to an endomorphism of T_p and since T_p±A, it can be further extended to an endomorphism of A.

Thus we have shown that the restrictions offto allp-components ofT are term functions. It is important to observe that there is a common term functiont which coincides with fon allp-components. Indeed, this follows from the fact that, by the induction hypothesis, all unary func- tions f( 0 ,R, 0 ,x, 0 ,R, 0 ) are term functions.

Our next step is to show that the restriction offtoTis a term func- tion. Since every element ofTbelongs to a sum of finitely manyp-compo- nents, we can use again an induction argument. Let

x1,R,xnTp₁5R5T_pm

and assume that the formula (3) holds if x₁,R,x_nT_p1 or x₁,R,x_n T_p25R5T_pm. Using the Chinese Remainder Theorem, we find integers k and l such that

(k1l)xi4xi, kxiTp₁, lx_iT_p25R5T_pm, i41 ,R,n. It remains to calculate:

f(x₁,R,x_n)4f( (k1l)x₁,R, (k1l)x_n) 4(k1l)f(x₁,R,x_n)

4kf(x₁,R,x_n)1lf(x₁,R,x_n) 4f(kx₁,R,kxn)1f(lx₁,R,lxn)

4a1(kx₁)1R1an(kx_n)1a1(lx₁)1R1an(lx_n) 4a1(k1l)x₁1R1an(k1l)x_n

4a1x₁1R1anx_n.

For the rest we assume, without loss of generality, that the restric- tion of f to T is zero. The next step is to prove the equality

f(a₁,R,a_n21,a_n1t)4f(a₁,R,a_n21,a_n) (4)

for all a1,R,anA and tT. Obviously, it is enough to consider the case tTp, for an arbitrary prime p. We treat two cases separately.

Case 1. T_p±A. LetA4B5T_p anda_i4b_i1t_i whereb_iB,t_iT_p, i41 ,R,n. Then, using Lemma 2.2, we have

f(a1,R,an21,an1t)4f(b11t1,R,bn211tn21,bn1tn1t) 4f(b1,R,bn)1f(t1,R,tn21,tn1t) 4f(b1,R,bn)1f(t1,R,tn)

4f(b11t1,R,bn1tn) 4f(a₁,R,a_n) .

Case 2.T_p±O A. Now we can find inT_pa cyclic direct summandCofA whose order is not less than the order oft. LetA4B5C,C4acb,a_i4 4b_i1c_iwhereb_iB,c_iC. Also, lett4u1vwhereuB,vC, and take the endomorphismfofAsuch thatfN^B41_Bandf(c)4u. Then calcula-

te, using again Lemma 2.2:

f(a₁,R,a_n21,a_n1t)4f(b₁1c₁,R,b_n211c_n21,b_n1u1c_n1v) 4f(b1,R,bn21,bn1u)1f(c1,R,cn21,cn1v) 4f(f(b1),R,f(bn21),f(bn1c) )

4f(f(b₁,R,b_n21,b_n1c) ) 4f(f(b₁,R,b_n)1f( 0 ,R, 0 ,c) ) 4f(f(b₁,R,b_n) )

4f(f(b₁),R,f(b_n) )

4f(b1,R,bn)1f(c1,R,cn) 4f(a1,R,an) .

So we have proved (4), from which it follows thatf(a1,R,an21,t)4 40 for arbitrary a₁,R,a_nA and tT. Indeed, (4) implies

f(a₁,R,a_n21,t)4f(a₁,R,a_n21, 0 ) .

Since by the induction hypothesis f(x1,R,x_n21, 0 ) is a term function and its restriction to the unbounded group T is zero, we must have f(a₁,R,a_n21, 0 )40 .

It remains to show that f(a₁,R,a_n)40 holds also in the case when none of theai is a torsion element. Since rank (A/T)41 , there are inte- gersk,land elementsaA,u,vTsuch thatan214ka1u,an4la1 1v. Then

f(a₁,R,a_n)4f(a₁,R,a_n22,ka1u,la1v)4f(a₁,R,a_n22,ka,la) . Again by the induction hypothesis, f(x₁,R,xn22,kxn21,lxn21) is a term function and its restriction to T is zero, so we conclude that f(a₁,R,a_n22,ka,la)40 . This completes the proof. r

REMARK. In some cases it may be useful to consider an abelian groupAas a module not overZbut over the largest subringNofQover which it is a module – this ringNis called thenucleusofA. For example, the nucleus of every divisible torsion-free group isQ. Clearly, ifNis the nucleus of a groupA andqN0Zthen the function xOqx is an endo- function but not a term function ofA. ThusAcannot be endoprimal as an abelian group. On the other hand, this function is a term function of the

N-moduleA, soAcan be endoprimal overN. Therefore, when investigat- ing endoprimality of abelian groups, it may be convenient to consider them as modules over their nuclei, that is, to regard the functions (1) withk_iNas term functions. It is easy to understand that the nucleus of a mixed groupAis the subring ofQgenerated by the inverses of all pri- mes p such that A/T is p-divisible and T_p40 .

For this case, as is easy to see, our main theorem takes the following form.

THEOREM 1.28. Let A be a group and P be the set of those primes p for which A/T is p-divisible. Assume A has torsion-free rank 1. Then A is endoprimal over its nucleus if and only if T is unbounded and, for every pP, at least one of the following three possibilities occurs: 1) T_p40; 2) T_p is not reduced; 3) T_p is not a direct summand of A. r

R E F E R E N C E S

[1] B. A. DAVEY- J. G. PITKETHLY,Endoprimal algebras, Algebra Universalis,38 (1997), pp. 266-288.

[2] P. C. EKLOF - M. HUBER, Abelian group extensions and the axiom of con- structibility, Comment. Math. Helv., 54 (1979), pp. 440-457.

[3] L. FUCHS, Infinite Abelian Groups, I-II, Academic Press, New York, 1970, 1973.

[4] R. GO¨BEL- K. KAARLI- L. MA´RKI- S. WALLUTIS,Endoprimal torsion-free se- parable abelian groups, J. Algebra Appl., to appear.

[5] K. KAARLI - L. MA´RKI, Endoprimal abelian groups, J. Austral. Math. Soc.

(Series A), 67 (1999), pp. 412-428.

[6] K. KAARLI- K. METSALU,On endoprimality of torsion-free abelian groups of rank 3, Acta Math. Hungar., to appear.

[7] L. STRU¨NGMANN - S. WALLUTIS, On the torsion groups in cotorsion classes.

In: Abelian Groups, Rings and Modules (Proc. Conf. Perth, 2000), Contempo- rary Mathematics,273 (2001), pp. 269-283. Amer. Math. Soc., Providence, RI.

Manoscritto pervenuto in redazione il 5 dicembre 2003.