Advances in Pure Mathematics, 2013, 3, 670679 Published Online November 2013 (http://www.scirp.org/journal/apm) http://dx.doi.org/10.4236/apm.2013.38090 Open Access APM The Lattice of Fully Invariant Subgroups of the Cotorsion Hull Tariel Kemoklidze Department of Mathematics, Akaki Tsereteli State University, Kutaisi, Georgia Email: kemoklidze@gmail.com Received September 9, 2013; revised September 18, 2013; accepted October 7, 2013 Copyright © 2013 Tariel Kemoklidze. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT The paper considers the lattice of fully invariant subgroups of the cotorsion hull T when a separable primary group is an arbitrary direct sum of torsioncomplete groups.The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and for mixed groups the ring of endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. In the considered case, the cotorsion hull is not fully transitive and hence it is necessary to introduce a new function which differs from an indicator and assigns an infinite matrix to each element of the cotorsion hull. The relation difined on the set T of these matrices is different from the relation proposed by the autor in the countable case and better discribes the lower semilattice . The use of the relation essentially simplifies the verification of the required properties. It is proved that the lattice of fully invariant subgroups of the group T is isomorphic to the lattice of filters of the lower semilattice . Keywords: Lattice of Fully Invariant Subgroups; Direct Sum of TorsionComplete Groups; Cotorsion Hull 1. Introduction We consider questions of the theory of abelian groups and throughout the paper the word “group” means an additively written abelian group. The notation and terminology used in the text are borrowed from the two volume monograph [1,2]. The symbol denotes a fixed prime number. p and are respectively the groups of integer and rational numbers. If is the order of an element of the group, then the exponent of an element is equal to and written as Qn p a a n ea n. A subgroup of the group B is called fully invariant if for any endomorphism of the group this subgroup is mapped into . The examples of such subgroups are B B AnA ana , 0, naa aAn , , , the torsion part of the group 0n nZ . The study of the lattice of fully invariant subgroups of a group is an important problem of the theory of abelian groups. For sufficiently wide classes of groups this topic is treated in [37] and in other papers. The works [813] and others are dedicated to the investigation of this question in torsionfree and mixed groups. p A group is called a cotorsion group if its ex tension by means of any torsionfree group splits, i.e., C ,CAExt 0 . The importance of the class of cotorsion groups in the theory of abelian groups is due to two facts (see [1, items 54, 55]): for any groups , , the group B Ext , B is a cotorsion group; any reduced group is isomorphically embeddable into the group Ext , QZA called the cotorsion hull of a group and, in addition, A is a divisible torsionfree group. Any reduced cotorsion group can be represented as the direct sum • TC where Ext ,TQZT , is the torsion part of a group =TtA and is a torsionfree, algebraically compact group (see [14]). If C p is represented as a direct sum of primary components, then TT Ext,Ext, . p QZTZ pT The construction of algebraically compact groups is well known (see [4, item 40]). Hence the study of cotorsion groups reduces the study of groups of the form
T. KEMOKLIDZE 671 Ext , pT where T is a primary group to a considerable extent. p Though the notion of a cotorsion group and its generalizations are studied sufficiently well (see [1518]), little is known about the lattice of fully invariant subgroups of a cotorsion group. The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and, as shown in [19], for mixed groups the ring of endo morphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. The study of the lattice of fully invariant subgroups makes essential use of the notions of an indicator and a fully transitive group. By the indicator of an element of the group p a we mean an increasing sequence of ordinal numbers ,,, , n A Ha Hahahpahpa , where denotes the generalized height of the element , i.e. hp a ha if and (certainly, if then ). In the set of indicators we can introduce the order 1 \ApA 0, na aph 0h 1n hp ahp ,0,1, ii HaHbhpa hpbi . A reduced group is called fully transitive if for its arbitrary elements and when p a ,b aHb there exists an endomorphism of the group such that ab . In fully transitive groups, the lattice of fully invariant subgroups is studied by means of indicators (see [2, Theorem 67.1]). A. Mader [11] showed that an algebraically compact group is fully transitive and described by means of indicators of the lattice of fully invariant subgroups of an algebraically compact group. Moreover, he indicated the generalized conditions the fulfillment of which gives a description of the lattice of fully invariant submodules. Theorem 1.1 (A. Mader). 1 Let be a module over a commutative ring , be the lattice of its fully invariant submodules, be some lower semilattice, and be the mapping with the following properties: R :A 1) is surjective; 2) aa ab and ; aA a b EndfA 3) ; 4) if then there exists an endo morphism of the module ,ab f such that ba ; 5) cC if , then for any there exists such that . C ,abC b ca * Then the set of all filters of , which is ordered with respect to the inclusion, is a lattice and the mapping * : defined by the rule AaD Da is a lattice isomorphism. In the same way as we did in groups we define the notion of full transitivity in the group p ,p •ExtTZ T T. If is a torsioncomplete group, then its cotorsion hull is an algebraically compact group (see [1, item 56] and, as has been mentioned above, is fully transitive. A. Moskalenko [13] proved that when is the direct sum of cyclic groups, then T T p is also fully transitive and all the conditions of Theorem 1.1 are fulfilled. Therefore in this case, too, the lattice * of indicator filters describes the lattice of fully invariant subgroups. The direct sum of torsioncomplete groups is a natural generalization of the direct sum of cyclic groups and torsioncomplete groups. The author has shown in [20] that in this class of groups, if the sum is infinite, the cotorsion hull is not fully transitive. Therefore, because of condition 4) of Theorem 1.1 we cannot use indicators to describe the lattice of fully invariant subgroups. The lattice of fully invariant subgroups of p T was studied in [21] when is the countable direct sum of torsioncomplete groups. In the present paper, is an arbitrary direct sum of torsion complete groups and the lower semilattice is defined by a simpler new relation T T (see Definition 2.2) which makes it easier to verify the properties of Theorem 1.1. 2. A Semilattice Let a group be the direct sum of torsion complete groups ppT , i i TB (2.1) where is a basic subgroup of the group i Bi B and i i BB pT is the basic subgroup of . Assume that T is a fully ordered set of indexes. For a separable group , A. Moskalenko [13] represented elements of the cotorsion hull T as countable sequences 01 1. ˆ ,,,, ,0,1, ii ii ,aa Ta TaT pa aTi T Writing the element in this form, it is easy to calculate their height and indicator (see [21], (1.2)) Let J Bx be a fixed basic subgroup of a separable group . If , , then in the group there exists a sequence pT BaT 01 ,,aaaT i b, 0, 1,i jj ii bm , such that for any i 1= ,0and lim. j sn s i iiis n js x mpapb 0 (2.2) This representation of an element is called a Open Access APM
T. KEMOKLIDZE 672 canonical. The sequence is said to correspond to the canonical representation of . The statements given below are true (see [13]). i b Ha a Proposition 2.1. If 01 T, ,,kk i b is the sequence corresponding to the canonical representation of an element , and between and 1i there is a jump, then in the expansion i ki b with respect to the basis a i kk x there is an element of order . 1 i k p Proposition 2.2. If T a is a sequence of nonnegative integers, then it has an infinite number of jumps. Let a group be of form (2.1) and TaT . Denote by i the projection of the group T on the direct summand i B and consider the sequence bb k ij ij , (see (2.2)). For each and fixed , the sequence defines the element 0,1,j1i , i k 01 ,, ii bb =0 ns s 2 ,, i Ta lim , ik s n apb (2.3) and the elements of the group define the element of the group 01 ,, ii aa 01 ii a ˆi B , i aa TT . It is obvious that 1 lim n k ni a . i k a (2.4) (Here we have separated two indexes i and ks by the comma and we will sometimes do so in the sequel in order to distinguish their order). Note that the elements , , are uniquely defined by an element (see [13], item 1.2) i a1, 2,ia To every element we put into correspondence the matrix aT 0i aH 0, i a (2.5) where T 00 aHa, ˆ 0 0ii T aHa for and t 1i he indicators are written in a column. Definition 2.1. The matrix ij k, , i0, 1,j , made up of ordinal numbers and symbols is called admissible with respect to the group if the following conditions are fulfilled: T 1) The 0th row 00 01 is an increasing sequence of ordinal numbers so that 0i ,, k kk or 0i k . If 0j k , then 01 0nn for any , whereas the other rows are increasing sequences of nonnegative integer numbers or symbols (Here 1kknj is the smallest infinite ordinal number and it is assumed that ). 1 2) If 0n km mn k is the first infinite ordinal number and , then infinitely many rows contain a nonnegative integer number and there exists a row such that ,inm for 0 If 0n 0 i , .iikm , then starting from some all rows consist only of symbols mn0 i . 3) If all elements in a row are nonnegative integers, then this row contains infinitely many jumps. 4) If between ij and ,1ij there is a jump, then in the group i there exists a bases element of order kk B 1 ij k p (it is assumed that ). 0 BB 5) In each column ij as (i.e. runs through all values of any fully completely ordered set ); also, if k m ii 0j k , then , and if 2 , jj kkk 0 min j 1 , 0j km , then . 12jj Taking into account equality (2.1) and Propositions 2.1, 2.2, we notice that the matrix satisfies the above conditions for any kk a aT . From Definition 2.1 it follows that we deal with matrices of the following three types: I. 00 01 10 11, kk kk where are nonnegative integer numbers or symbols ij k ; II. 000 1 101 1 1 1 , n n tn km m k k 01 11 01tt kk kk kk n where m and ij are nonnegative integer numbers (see the first sentence in item 2 of Definition 2.1); k III. 0000 1 101 1 10 1 1 . n n ii ini kkknk nk kk k kk k 1 11 11 Here ij are nonnegative integer numbers (see the second sentence in item 2 of Definition 2.1). k Note that if 01 ,,aaaT T is given and i b ab is the sequence corresponding to a canonical representation of an element then 00 1 2 ,a 2 pbpb b and each i is the sum of finitely many bases elements (see (2.2)). Hence it follows that in any matrix , at most a countable number of rows is different from . Moreover, by virtue of the fourth condition of Definition 2.1, from the given ith row of an admissible matrix it is easy to find an element 0 a ,, T ˆ i a the indicator of which is equal to this given row. Let for example Open Access APM
T. KEMOKLIDZE 673 1011 12 101112131415161718191,101,11 ,,, 1,2,3,5,6,9,10,11,12,15,17,18, kkk (2.6) (it is obvious that here 10 11 and the indexes are marked in order to determine in which row and column an element lies). Jumps here occur at positions 12 13 3,5 a , , , . Then, by virtue of the fourth condition of Definition 2.1 (see also Proposition 2.1), in the canonical representation of 14 15 6,9 18 19 12 ,15 19 1,10 15 ,17 b 10 there is the element 12 1,3 211b containing a basis element of order . Just in the same way contains a basic element of order ; contains an element of order ; an element of order , an element of order and so on. Therefore 31 4 pp 4 b 16 b 17 b 14 1,6 412 b 61 7 pp 12 113 pp 15 116 pp 18 119 pp b 18 1,12 8 b 19 1,15 b 1,11 1,18 b 1 9 11 2467 101112 14 16 17.a pbpbpbpbpb The indicator 10 a is obviously equal to sequence (2.6) and 0i is not the only element the indicator of which is equal to (2.6). a Denote by the set of admissible (with respect to ) matrices and define, on the set the following relation different from the relation given in [21, Definition 1.2]. T Definition 2.2. Let ij k, ij k , i , , be the elements of the set . We say that 0,1,j K ij k 1i if , , and to each element where there occurs a jump we can put into correspondence the element where there also occurs a jump so that and nj . Then the following two conditions are fulfilled: 00ii kk 0, 1,i mn k mn m j 1 ij k kn 1) Each element mn where there is a jump has finitely many (possibly none) preimages. k 2) If 01 , are infinitely many elements of the ith row of the matrix , ij ij kk which are different from the symbol and 0011 mn are respectively their preimages such that the sequence of the numbers of rows infinitely increases, then 01 ,,mm ,, mn kk kk jn as . k It can be easily verified that the relation on the set is reflexive and transitive. However, as seen from the next simple example, it is not antisymmetric. Indeed, let , ij Uu , ij Vv , be admissible ,i0,1 ,j matrices all rows of which, except for the third one, are identical. Consider the table j 0123 4 5 6 7 8 910 00jj uv1235 7 8 9 10 121415 11j uv j 123 679 10 12 131415 22jj uv2345 7 8 9 10 121415 3j u 4 5 68910 12 13 14 16 17 3j v 5 6 9 10 12 13 17 18 19 21 22 j 11121314 15 16 17 18 00j uvj j 171920 25 26 27 29 30 11j uv 181920 25 26 27 29 30 22j uvj 172021 26 27 28 30 35 3j u 182324 27 28 30 31 36 3j v 232728 29 31 32 37 38 Let us assume that the elements lying at the positions of dots in the matrices and V are identical. To the elements of the third row of the matrix , where there are jumps, we put into correspondence the elements of the second row of the matrix . Thus UU V 3223 352738283,112,11 3,132,133,152,17 3,172,17 65,1010, 1412, 1817, 2421 ,2830,3130 . To the elements of the third row of the matrix V where there are jumps there correspond the elements of the third row of the matrix in the following manner: U 31323335 3535 383,11 3,113,133,14 3,173,163,17 66, 1010, 1310, 1918, 2324,29 31,32 31. It is obvious that UV and , whereas VU UV . Therefore the relation on the set is not antisymmetric. Then and def UVU VVU is the relation of equivalence on the set , whereas def UV UV defined on the factor set is the relation of order. Let ij Uu, ij Vv, , , be admissible matrices. We denote i0,1,j min , ij ij WUVuv ij w and will show that is also an admissible matrix. Let W 01 ,,, ii uu (2.7) 01 ,, ii vv (2.8) Open Access APM
T. KEMOKLIDZE 674 be respectively the ith rows of the matrices and V where there occur infinitely many jumps. Let us show that then U 00 11 min,,min , , ii ii uvuv (2.9) is also an increasing sequence of nonnegative integer numbers where there are infinitely many jumps. Assume the contrary: let, starting from some number in (2.9), there are no jumps and isisis . Assume that is the first jump to the right from is in (2.7), and is the first jump to the right from in sequence (2.8) and . Then ij , ,1 , wuv jm 1ij ij wu ,1 , iji j uu is v ,1is is wu u ,1 , imim vv ,1 ,,wij u and, obviously, . If , then ,1,1im im and ,1 ,1imim , which contradicts the definition of W. Therefore the third condition of Definition 2.1 is fulfilled. ,1ij ij wv w,1 vjmwu Let be a jump in (2.9) and ,1 , iji j ww ij ij wu . Then ,1,1 and is a jump in (2.7). Then ,1,1 ij ij iuiw , i.e. the fourth requirement of Definition 2.1 is also fulfilled. The fulfillment of the remaining conditions of Definition 2.1 is obvious. Therefore is an admissible matrix. 1 ij ij w u BB W 1 ij ij uw ,1i j , ij uu 0 It is not difficult to verify that W and WUV . Moreover, if ij kU and , V then . W Let now ,UV, where U and V are admissible matrices. Let us define the exact lower bound of U and Vas follows: inf ,UVU V W where min ,, ij ij Wuv , i 0,1,.j If ij UuU and ij VvV , then, by virtue of the above properties, min , ij ij WuvU U and Hence . WVV WW and, by symmetry, , i.e. WW =WW , which shows that the definition of =WUV is reasonable. Since W, , we have UWVWU , WV, and if U, V , we have U , V, W, i.e. W. Thus all conditions of the definition of the exact lower bound are fulfilled. Therefore the set with relation is the lower semilattice. 3. The Lattice of Fully Invariant Subgroups of the Group T Let us show that the function :,T aa, where has form (2.1) and T is the set of all admissible matrices with respect to , satisfies all conditions of Theorem 1.1. T Condition 1. is surjective. Proof. Let , =ij k, , i0,1,,j 1 ,kk and the 0th row consist of nonnegative integer numbers. For any th row, denote all jumps by ss ii ii, . By virtue of the admissibility of the matrix , for each jump of this kind we can choose in the basis 1i 1,2,s K ii x of the group an element i Bis of order 1 s ii k p . Let s ii sis bi 12 12 ii i xpx . Denote , then is i is x is cp p0lim ii s ac s ; 0 aˆ ii B . Taking into account that and for every 1 1ii 10i k i ki , 1111 ss ii s k 1,i is ii s ki we see that ˆ1 ii B Ha 12 00 ,, ii kk. Further, since ii tii it and the matrix is admissible, for each fixed we have tt ki li s as . Now we can define the element , i 00 ˆ i 1 ms i aa T 00001 ,,k k ˆ T Ha0 a and . Since ˆ TT is a divisible group, there are elements 12ˆ aa T ,, such that for any 0,1,i we have . Let 1ii pa TT a ,T 01 ,a aa , then aT and aK , aK . Now assume that the matrix is of type II. Then its 0th row has the form K 00010 1 ,,, ,,1, n kkkm m where mn and there exists an index 0 i such that in m k i 01 x for every 0. Like in the preceding case, for each th row there exists an element 2i iiiisi Here the number of summands is finite since the row has finitely many jumps. Now, taking into account that 12 ii 1 12ii p 1. is x pxB ap i iii s and the matrix is admissible, we obtain t itii t K ki as , . Then we can define the element i 1 t in 12 012 1 ˆ lim js jj j js j k jj kj apxpxpx T 1 j n and ˆ0000101 ,,, ,, n B Ha kkk . Denote 12 12 1 lim js jj j js tk kk kjj tj apxpxp x assuming that that when 0 jr k jr px r l . Then it is obvious that 1kk a T pa , . Consider an element 0,1,k ,,aT T 01 aa . It easily follows that aK . Therefore aK . If is a matrix of form III, then, starting from some th row, every row consists only of symbols K t . We choose a row 01 ,, ii kk is ii i is px , , and, just in the same way as in the preceding case, find , 1it ik i 1 ap 01ii x B, , and 1, 2,,i ks , 00 iii Bk1 ,,,, iin k k 1 Ha . Let us consider an Open Access APM
T. KEMOKLIDZE 675 element 1 1 01 1 js jj js j t j j apx px T, 0n ap, 1 j n . Then . Since 0000101 ,,, ,, Tn Ha kkk ˆ TT is a divisible group and , there exists in ˆ TtTˆ TT a quasicyclic divisible subgroup. Let 1, 2, i pg . ii pgg T T i gTi be its system of generators such that 1 and for every 1Since is a pure subgroup in , it can be assumed that T i ,i ˆ T p 0, .12 k aa a for each Now let 1, 2,iki i ag and consider the element . It is obvious that , i.e. 01 1 ,,,, , kk aT aTaT K aa a aK. Condition 1 is proved. Condition 2. If and aT End , T then af a. Proof. Let , aa 01 ,,aT 01 ,,cccT T and there exist an endomorphism of the group T such that ac . As is known (see [22, item 1.5]), is induced by the endomorphism of the group T which in its turn induces an endomorphism ˆ of the algebraically compact group such that ˆ T 01 01 ˆˆ ,, ,, afafaTccT c. Let and i b i d c be respectively the sequences corresponding to canonical representations of the elements and . Then (see (2.3), (2.4)) a 000 0 ,lim ns ii i n is aaa pb , s 0j where runs through at most a countable set of increasing indexes. Since is an algebraically compact group, we have iˆ T 000 ˆˆ. i ij afacc (3.1) Let , ij ak , ij ck ,i0,1,j ij k, and assume that there is a jump at the position , . Then in a component of the element there 1i 0 c 0i c kj exists an element , ij ij ik j pd of the exponent 1j which, by virtue of equality (3.1), is obtained by mapping the sum of a finite number of summands of the element 0 under the homomorphism aˆ and by projecting this mapping on a basis element of the subgroup i that has the exponent . Denoting this projection by B1 ij k ,1 ij ik , we have 1 11 11 11 ,1 ,, , ˆ , ijs ij ss iji jsijs ss ij ij kj ik i kjikj kj ik j fp bpb pd j kj (3.2) where the abovementioned sum of a finite number of summands is enclosed in the brackets. It is obvious that the height of such a summand is less than or equal to ij k , whereas the exponent is greater than or equal to 1k . Hence without loss of generality we put into correspondence to ij k an element s ij of the largest exponent. Thus, to each element ij of the matrix k k where there is a jump we put into correspondence an element of the matrix . K Let 1 be elements of a row of the matrix where there are jumps, and 00 11 be respectively their preimages for the abovementioned correspondence so that a sequence of numbers of the rows 01 of the matrix 0 ,, ij ij kk K ,,mm 1i ,, mn mn kk increases infinitely. Taking into account that is an algebraically compact group and its torsion part is a torsioncomplete group, by virtue of equality (3.2) we assume without loss of generality that ˆ T ,1 ,, ˆ,0,1, ijtmnt ttt tmnij ttmn t tt ttt kjk n mkik jmkn fp bpdt . We have 10 110 0 01 01 01 01 01 00 11 00 11 00 11 ,1, 1 ,, ,, ˆ . mnm n ij ij ij ij mn mn mn mn mk mk kj kj ikjikj kn kn mknmk n fpbp b pd pd (3.3) Since ˆ induces an endomorphism on and ,T , tmn tt mk 1 are projections, 10 110 0 ,1, 1 ˆ ˆmnmn mk mk induces an endomorphism on the subgroup . Let us fix a positive integer number and consider an element of order from T: T m m p 01 01 01 1 * 0, , . ij ij ij ij km ik jikj apbp b 1km If here the initial summands s ij , then we assume that these summands are zero. Then, in view of (3.3), 10km 00 11 01 01 01 00 11 00 11 00 11 00 11 11 * 0, , 11 ,, ˆˆ . ij ij ij ij mn mn mn mn kjjm kjjm ikjik j knjm knjm mknmk n apbpb pdpd (3.4) Since, by condition, a sequence of numbers of the rows 01 of the matrix ,,mm increases infinitely, only a finite number of summands on the righthand part of equality (3.4) must differ from zero; otherwise the element * 0 ˆa does not belong to . Therefore for each concrete positive integer , starting from some we have T mt 0m tt jn , tt jnm , i.e. tt as . Thus the second condition of Definition 2.2 is fulfilled too. Therefore jn t aaf and Open Access APM
T. KEMOKLIDZE 676 af a. Condition 2 is proved. Condition 3. For any ,ac T , ac a c Proof. Let , be the elements of the group 01 ,,aaaT T 01 ,,cccT . Then (see [13, item 1]) . Denote (see (2.5)) 0011 ,,cac T ac a 0,a i aH 0, i ccH , ij ack 00iiij acHa ck , i, . By virtue of the properties of the indicator (see [1, item 37]), 0,1,j 010 000 01 ,, ,, iii iii ii kkHa cHaHc kk s for any i. Let and at is k there be a jump, then isis . If to the right from the first jump occurs at the position ist , then is and kk 1i k k is is t kstk ts . To the element we put into correspondence the element For this correspondence, if 12 are the elements of the ith row where there occur jumps, then their preimages lie in the same ith row of the matrix is is t k k. , is is kk, ij k and therefore the second condition of Definition 2.2 will be fulfilled, too, i.e. ac c a or c ,ac aca. Condition 3 is proved. Condition 4. If and T ac , then there exists an endomorphism of the group T such that . ac Proof. Let , and 01 aa nn ,,aT 01 ,,cccT 00 lim,lim,0,1, , ss ii si is nn ss apbcpdi where i b, i d are respectively the sequences corresponding to their canonical representations. 1) Denote ij ak, ij ck , i ,,kk , , and assume that the sequence 00 01 contains only nonnegative integer numbers. Let 12 be the elements of the irow which have preimages in the matrix . Denote the preimages of the element 0,1,j ,, ij ij kk c ij k as follows 11 22 ,,,,1,2, ssssss ms ms ininin kk ks . (3.5) Then the element 0i has the summand ,ij s s, where contains, as a summand, the basis element a ij s s kj ik j pb ,ijs s ik j b si x of the exponent 1 s ij k . Denote ,1,2,, ,. ss mm isi s sij smm sin Axis kjk n We will show that for each , 1 1 12 11 22 1 ,1, 1 ss ss s s ss s ms ss sssss s msms ii ii ii bpbp b pdpdpd i (3.6) and 0 ix for any basic element contained in the sequence i b of the canonical representation of when a i A . Note that when 1 , the summands of the element 21 11 1 ,1, 1iii bpbpb 2 do not contain the basis element when . 2s Indeed, when 21 ,t 21 2ij ij j we have 12 11 1tk kj . Hence 1 11 , 1 ij tj k 1 . (3.7) On the other hand, 11 ˆ0, jiij T hpak 0i a 11 and, if we take into account the definition of the height and the representation of the element , then for each 1 ,1ij nk we will have . Therefore, by (3.7), 1 ,1 0 mim j pb 11 11 1 11 1 11 , ,11 0, tjtj it itj j pb pb but for each 1 2 2 1 12121 21 11 11 s ij ijij s tjjkj j kjjkex 1 . Denote by t m the coefficient with which is contained in the expansion of the element . Recall that t b 0t s mp . The condition 21 11 2 112 21 1 11 1 1 11 12 11111 1 11 2211 1 ,1, 1 11 1 111 1 m mm ii ii i ii bpbpb mpmpm x pdpdp d i (3.8) must be fulfilled for the sought homomorphism i . Since 1 participates in the expansion 1 i b with respect to the basis, we have 1 1,mp 1. Therefore 1i can be uniquely defined in the subgroup from (3.8) if B 1 11. iij ex k (3.9) But this inequality holds true because the preimages of 1 ij are (3.5) (for k1 ) and, by the definition of the relation between the matrices and a c, for each , 1m1, 2,t we have 1 11 1 11 1 111 11 1 1 11 11 1 1 1 0, ij ij t in ij ttt tt tt t in tt tt kk nkj k ii kjn i ppd pd pd 1 ij there exists an endomorphism i of the group such that T Open Access APM
T. KEMOKLIDZE 677 since , which proves the validity of inequality (3.9). 1 1t jn Just in the same way as in the case 1 we will show that in the expansion of elements 22 3 ,1 ,1 there is no ,,, iii bb b when . Therefore the condition 2s 32 223 3 22 32 3 22 32 2 22 2 2 22 12 222222 11 2222 1 ,1 ,1 1 11 11 11 1 11 22 22 m mm ii ii i i ii bpbp b mpmpm x mpmpm x pdpdp d i is fulfilled for the endomorphism . i Since , according to this condition we define 2 2,mp 2i 1 uniquely. Just like for 1i we can verify that and so on. i exk 2 2ij The endomorphism i 1 is likewise defined uniquely by giving the images of basis elements and it is obvious that i maps the basic subgroup into. The endo morphism i uniquely continues up to the endo morphism ˆi of the group . Let us show that ˆ Tˆi induces an endomorphism on the group . T Let . Since the group has form (2.1), it suffices to show that tTT ˆitT when i tB. Since for the basis elements we have i ˆ0 x when i A, using (3.6) let, without loss of generality, be an element of order , t 1 m p 1 1 11 21 11 2 1 1 1 1 1 1 1 1 1. ij s ijs sss n km n ii i n km n ii i tpb pbpb pbpbp b 1 1 s s 0 (Here it is assumed that , and if in , s j s n ij ep bm several summands s ij , then these summands are equated to zero.) Then 1km 1 1 11 11 1 11 21 2 1 1 11 1 11 11 112 11 11 11 22 11 ,1 11 ,1 1 1 ,1, 1 1 1,, ij ij ij ijs s sij ij s sss ss ij km nkm iii i nkm km iii nkm nk m ii km ii tpb pb pbpb pb pb qp pd pd 11 11 11 11 11 2 22 1 , ,, , s mij s sss mm s ss ms ssss s ms ms km s i ii pdqp pd pdpd ,i (3.10) where , i qp 0 . If in equality (3.10) the numbers of rows 12 ,,, ss ii ims , 1, 2,s , of the matrix ()c infinitely increase, by virtue of Definition 2.2 (see also (3.5)) j sms jn when infinitely increases. But then 1 1 1 '1 . ss s msms ss msms ss ss msmsms ms s ij s ms s s ijij s ms sin s sms in in i km kmk nk kjmn ked j This means that starting from some all summands on the righthand side of equality (3.10) are equal to zero. Therefore ˆitT . The sum of endomorphisms ˆi i ˆ ˆ , which, on the algebraically compact group T, is induced by the endomorphism i on the ith component of the group i i T B , is the endomorphism of the group which ˆ T maps the subgroup into. It can be easily verified that 00 T ˆac . ˆ in turn defines the endomorphism of the group T for which • 01 01 ˆˆ ,, ,,aaaT ccT c At the beginning of the proof we have assumed that the 0th row 00 01 ,,Ha kk T consists of non negative integer numbers and in the ith row there are an infinite number of jumps. It is obvious that this reasoning is also true when the ith row contains a finite number of jumps (at least one jump) or when the 0th row of the matrix a has the form 000101 ,,, ,,1, n T Ha kkkmm and 0,,,ccTT, 0 cT . 2) Let us separately consider the case where ,1,Ham m m aa a T or, since in is pure, assume that in the notation we have 01 T 01 ˆ T ,,aaaT 0 and 1m,aT 1. m ap s Then in the representation 1 0 lim , i m si apb 1m i contains a basis element n b n nj B tt of arbitrarily large order and takes values from an infinite set of indexes. Let us fix a sequence 12 n n t j of positive integer numbers such that the expansion of contains a basis element n t b n n B such that nn .extm Let 1, 2, n Axn. We assume that 0x for A and, analogously to part 1 of the proof, for any we define nn such that the equalities Open Access APM
T. KEMOKLIDZE 678 1 1 1 11 1 1 1 1 nn nn n nn nn n tt ttt tt tt t bpbpb dpdpd 1 (3.11) are fulfilled. It can be assumed that 1. Note that 1tm nk n ts b0, tm , , does not participate in the expansion with respect to the basis when 1n . Indeed, if it were not so, then the exponent of the element 1 11 nn n t would be larger than 1, which is not so, 1, 2,k 1,, n st nn tm pbp 1t 1 tt b 2 1n t m p b 11 m ea . Moreover, n 1 2 n tm t 1 m nn exex. Indeed, when , a coprime number with respect to serves as the coefficient 1n p1 . Therefore it suffices to show that . We have 11 tt pd 12 2 1 11 tt t pd 1 0 tm pd 1 1 2 2 1 11 2 2 1, tm mm mt tm t cdpd pd pd but 1 1 2 2 2 112 1 1 0 . tm mmmt tm t pcpdpdpd pd Hence, since mi is not divisible by , all summands must be equal to zero, i.e. 11 . Analogously, 2 and so on. It is obvious that by virtue of (3.11), n d 2 x p x ex ex e T and, by our construction, n belongs to various n B and therefore induces the endo morphism on the group and maps the subgroup into. For the induced endomorphism ˆ T T on the group we have ˆ T 1 1 0,0,,0,,,, 0, ,,,. mm m aTaTTaT Tc Tc 1 3) Let . Then 00 010 ,,,,,1, n T Ha kkkmm ,1, nn TT pammH pc and, as shown in part 2 of the proof, there exists an endomorphism of the group which induces the endomorphism on the T so that T nn pa pc . Hence , i.e. 0 n pf ca actT and TT T tH acf Ha . Therefore, by virtue the last sentence at the end of part 1 of the proof, there exists an endomorphism of the group T which induces the endomorphism on the subgroup so that . Then T a t ac a or ac . Obviously, f a C c induces the endomorphism on and . Condition 4 is proved. TCondition 5. If is a fully invariant subgroup of the group and , then there exists T,abCcC such that ca b . Proof. Let and be respectively the ith rows of the matrices 01 ,,kk 01 ,,ll a and b, . Let be a smallest index such that 1ij j kl . In these sequences, the nearest jump is to the right from . Let this jump occur at the position j 1 in the sequence , ss kk . In the latter sequence, to the left from there is the preceding jump. Let this jump occur at the position , then, from 1t j 1 , tt kk k to k inclusive, we add to each element so that between s m km and 1 k there would be no jump. Obviously, 1,, ts kmkm l l (3.12) exceeds respectively 1ts and, to the left from the index ,, , each ii kl . The obtained sequences 1 and 1 , where 1 and differ from 1 by the elements of (3.12), obviously satisfy the conditions of a row of the admissible matrix and 11 . Now let us assume that in the sequences 1 and 1 the equality of elements takes place at the number , nn n kl , where . Then, in these sequences we have, to the right from , a jump and repeat the previous reasoning. If in the sequences there are infinitely many jumps and at each stage the first jump occurs in one and the same sequence r nsj n , then not to violate condition 3 of Definition 2.1 we proceed as follows: let in the sequences r and r , at the position , n n n lk and to the right from there occur a jump between n 1 ,ll mm. Then in the first sequence r , where there are infinitely many jumps, there exists a jump 1 , ss kk such that t kl l n l ,, . On the right from t, we increase the numbers 1tm so that there would be no jump between the numbers m, 1m ll l l , i.e. in the sequence r we ourselves have intentionally created a jump between the numbers t and 1t l l . Note that at this position the condition of admissibility of a row has not been violated since ts , and there exists a jump between lk k and We have 1. s k 12 ; 12 . Denote i * , i * , 1, 2,i . The rows * and * are admissible and ** , where each element of * differs from the corresponding element of * . Now, if in every row of the matrices and a b we perform such transformations, then we obtain admissible matrices and the corresponding elements of which differ from one another and U V aU, bV , abUV . It is not difficult to verify that this reasoning holds for all type of matrices a and b. Since U and are admissible matrices, there exists V , yT such that U and V . Then, by virtue of condition 4, there exists • ,End T such that , ax by . Hence , yC , , yab yc C. This means that ca . b Therefore bca. Condition 5 is proved. Open Access APM
T. KEMOKLIDZE Open Access APM 679 We have obtained that the function :T , where has form (2.1) and the set of all admissible (with respect to ) matrices satisfies the conditions of Theorem 1.1. Hence the following statement is true. TT Theorem 3.1. The lattice of fully invariant subgroups of the cotorsion hull of a direct sum of torsioncomplete groups is isomorphic to the lattice of filters of the semilattice p. 4. Acknowledgements This study was supported by the grant (ATSU2013/44) of Akaki Tsereteli State University. REFERENCES [1] L. Fuchs, “Infinite Abelian Groups. I,” Academic Press, New York, London, 1970. [2] L. Fuchs, “Infinite Abelian Groups. II,” Academic Press, New York, London, 1973. [3] R. Baer, “Type of Elements and Characteristic Subgroups of Abelian Groups,” Proceedings London Mathematical Society, Vol. s239, No. 1, 1935, pp. 481514. http://dx.doi.org/10.1112/plms/s239.1.481 [4] I. Kaplansky, “Infinite Abelian Groups,” The University of Michigan Press, Ann Arbor, 1969. [5] R. S. Linton, “On Fully Invariant Subgroups of Primary Abelian Groups,” Michigan Mathematical Journal, Vol. 22, No. 3, 1976, pp. 281284. http://dx.doi.org/10.1307/mmj/1029001528 [6] J. D. Moore and E. J. Hewett, “On Fully Invariant Sub groups of Abelian pGroups,” Commentarii Mathematici Universitatis Sancti Pauli, Vol. 20, 19711972, pp. 97 106. [7] R. S. Pierce, “Homomorphisms of Primary Abelian Groups,” In: Topics in Abelian Groups (Proc. Sympos., New Mex ico State Univ., 1962), Scott, Foresman and Co., Chicago, 1963, pp. 215310. [8] A. R. Chekhlov, “On Projective Invariant Subgroups of Abelian Groups,” Vestnik Tomskogo Gosudarstvennogo Universiteta Matematika i Mekhanika, Vol. 2009, No. 1, 2009, pp. 3136 (in Russian). [9] R. Göbel, “The Characteristic Subgroups of the Baer Specker Group,” Mathematische Zeitschrift, Vol. 140, No. 3, 1974, pp. 289292. http://dx.doi.org/10.1007/BF01214169 [10] S. Ya. Grinshpon and P. A. Krylov, “Fully invariant Sub groups, Full Transitivity, and Homomorphism Groups of Abelian Groups. Algebra,” Journal of Mathematical Sci ences (New York), Vol. 128, No. 3, 2005, pp. 28942997. http://dx.doi.org/10.1007/s1095800502455 [11] A. Mader, “The Fully Invariant Subgroups of Reduced Algebraically Compact Groups,” Publicationes Mathe maticae Debrecen, Vol. 17, 1970, pp. 299306. [12] V. M. Misyakov, “On Full Transitivity of Reduced Abe lian Groups,” In: Abelian Groups and Modules, No. 11, 12 (Russian), Tomsk State University, Tomsk, 1994, pp. 134156 (in Russian). [13] A. I. Moskalenko, “Cotorsion Hull of a Separable Group,” Algebra i Logika, Vol. 28, No. 2, 1989, pp. 207226 (in Russian); Translation in Algebra and Logic, Vol. 28, No. 2, 1989, pp. 139151 (1990). [14] D. K. Harrison, “Infinite Abelian Groups and Homologi cal Methods,” Annals of Mathematics, Vol. 69, No. 2, 1959, pp. 366391. http://dx.doi.org/10.2307/1970188 [15] S. Bazzoni and L. Salce, “An Independence Result on Cotorsion Theories over Valuation Domains,” Journal of Algebra, Vol. 243, No. 1, 2001, pp. 294320. http://dx.doi.org/10.1006/jabr.2001.8800 [16] S. Bazzoni and J. Štóviček, “SigmaCotorsion Modules over Valuation Domains,” Forum Mathematicum, Vol. 21, No. 5, 2009, pp. 893920. http://dx.doi.org/10.1515/FORUM.2009.044 [17] R. Göbel, S. Shelah and S. L. Wallutis, “On the Lattice of Cotorsion Theories,” Journal of Algebra, Vol. 238, No, 1, 2001, pp. 292313. http://dx.doi.org/10.1006/jabr.2000.8619 [18] M. Hovey, “Cotorsion Pairs, Model Category Structures, and Representation Theory,” Mathematische Zeitschrift, Vol. 241, No. 3, 2002, pp. 553592. http://dx.doi.org/10.1007/s0020900204319 [19] W. May and E. Toubassi, “Endomorphisms of Abelian Groups and the Theorem of Baer and Kaplansky,” Jour nal of Algebra, Vol. 43, No. 1, 1976, pp. 113. http://dx.doi.org/10.1016/00218693(76)901393 [20] T. Kemoklidze, “On the Full Transitivity of a Cotorsion Hull,” Georgian Mathematical Journal, Vol. 13, No. 1, 2006, pp.7984. [21] T. Kemoklidze, “The Lattice of Fully Invariant Sub groups of a Cotorsion Hull,” Georgian Mathematical Journal, Vol. 16, No. 1, 2009, pp. 89104. [22] T. Kemoklidze, “On the Full Transitivity and Fully In variant Subgroups of Cotorsion Hulls of Separable p Groups,” Journal of Mathematical Sciences (New York), Vol. 155, No. 5, 2008, pp. 748786. http://dx.doi.org/10.1007/s109580089240y
