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saqarT velos mecn ierebaTa erovnuli ak ad em iis moam be, t. 4, #1, 20 10 B UL L ET IN OF THE GEORGIAN NAT IONAL ACADE MY OF SCIENCE S, vol.

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1, 20 10 Mathematics The Lattice of Fully Invariant Submodules of a Reduced Cotorsion p-adic Module Tariel Kemoklidze A. Tsereteli State University, Kutaisi (Presented by Academy Member H. Inassaridze) ABSTRACT.
The paper considers the lattice of fully invariant submodules of a reduced cotorsion p-adic module • T ⊕ C, where T is a countable direct sum of torsion-complete p-groups, and C is a torsion-free, algebraically compact group.

It is shown that this lattice is isomorphic to the lattice of filters of a semilattice made up of infinite matrices and indicators.

© 2010 Bull.

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Key words: cotorsion group, torsion-complete p-group, fully invariant submodule.
The paper deals with questions of the Abelian group theory and the term group always means an additively written Abelian group.

The notation and terms used in the paper are taken from [1, 2].

We denote by p a fixed prime number.

Z and Q are respectively the group of integers and the group of rational numbers.

In the p-adic topology, the completion of a torsion p -group T is denoted by the symbol Tˆ .

Its torsion part T is a torsion-complete group.


A subgroup B of a group A is called fully invariant if every endomorphism of A maps B into B .

Such are, for example, subgroups nA = na { ∈ Aa }, = {][ a nA na = ,0 or the torsion part of a group A .

∈ nAa }, ≥ ,0 ∈ Zn , The investigation of the lattice of fully invariant subgroups of a group is an important task of the theory of Abelian groups.
For a sufficiently wide class of p -groups this topic was studied by R. Baer [3], I. Kaplansky [4], P. Linton [5], R.Pierce [6], J. Moore and E. Hewett [7] and other authors.

For torsion-free and mixed groups, problems related to this topic were investigated in the works of A. Mader [8], R. Göbel [9], A. Moskalenko [10], S. Grinshpon, P. Krylov [11], V. Misyakov [12] and other authors.

Little is known about the lattice of fully invariant subgroups in the class of cotorsion groups.

A group A is .
A group A is .

The importance of is a called a cotorsion group if its extension by any torsion-free group C splits, i.e.

the class of cotorsion groups rests on the following two facts: for any groups A and B the group =ACExt , ( BAExt 0 ) ( ) , cotorsion group; any reduced group A is isomorphically embeddable into the group • A = AZQExt / ) ( , called the cotorsion hull of A ; also, represented as a direct sum ( while TAZQExt / C ≅ ) / , • A A/ = • TA is a torsion–free divisible group.
Any reduced cotorsion group A can be ⊕ is the torsion part of the group A , is a torsion-free, algebraically compact group.

If the group T is written as a direct sum TZQExt / , where T ≅ T = tA C , ) ( , • © 2010 Bull.

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18 of primary ( / TAZQExt / , ) components ( ) ( , pZExt ∞ / TA ∏≅ p T ⊕= p ) ∏= p C T p , then we have and p ( / TZQExt , Tariel Kemoklidze ( ( pZExt ∞ ) , ) T p ) ∏≅ p T • p , ∏= p • ∏= T ⊕ ( p p Thus the study of reduced cotorsion groups is reduced to a considerable extent to the study of groups of the form = • TA A ⊕ C C , . )

p where T is a p -group, and C is a torsion-free, algebraically compact group.
As it is shown in [1, §§40, 54], and therefore A too, are p -adic modules.

(1) • CT , When studying the lattice of fully invariant subgroups of a group A , active use is made of the notions of an element indicator and a fully transitive group.
By the p -indicator of an element a of a group A we mean the increasing sequence of ordinal numbers aH )( A ≡ aH )( = pahah ), (( ( ), ... n aph ( ),...), where h denotes the generalized p -height of a , i.e., σ=)(ah if a ∈ σ pAp \ +σ 1 A and ∞=)0(h (of course, if n aph ( ) = h )0( ∞= , then ph ( n + 1 = a )0) .
In a set of indicators we can introduce the order ( i bph i aph ( )( bH ⇔ ) , = ≤ ≤ ) i )( aH ,...1,0 ≤ A reduced p -group is called fully transitive if, assuming that .ba =ϕ , for arbitrary elements a and b )( aH there exists an endomorphism ϕ of the group such that In fully transitive groups, the lattice of fully invariant subgroups is studied with the aid of indicators.
In particular, I. Kaplansky showed that such a subgroup has the form )( bH uA )( ∈= { aHAa )( ≥ u }, o u σσσ= ,...) , n σ and ( ,... where condition: if between Theorem 67.1]).
n 1 is an increasing sequence of ordinal numbers and symbols ∞ that satisfies the following σ (see [2, σ there is a jump, then in A there are an element of order p and height 1+n n A. Mader [8] 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.

He also indicated the generalized conditions, the fulfillment of which gives a description of the lattice of fully invariant submodules.
Theorem 1.

(A. Mader).

Let A be a module over a commutative ring R , Δ be the lattice of all its fully invariant submodules, Ω be some lower semilattice, and Ω→Φ A: : be the map with the following properties: )( a 1) Φ is surjective; fa Φ≥ Φ 2) ( ) +Φ 3) ( ) ba a Φ≥Φ )( Δ∈C Aa ∈∀ ( b and f ∈ EndA ; ); Φ∧Φ≥ )( a , then there exists an endomorphism f of the module A such that )( b .
)( b 4) if Cc ∈ such that 5) if Then the set Ω∗ of all filters of Ω , which is ordered with respect to the inclusion, is a lattice and the mapping Da } )( Δ→ defined by the rule is a lattice isomorphism.

∈Φ∈α { Φ∧Φ=Φ , then for any )Dα = ( there exists Cba ∈, )( bf )( a )( c A = a ; α: Ω∗ Like in the case of p -groups, we will formulate the notion of full transitivity in the group • T = pZExt ( ( ∞ ), T ).
If T is a torsion-complete group, then its cotorsion hull is an algebraically compact group (see [1, §56]) and, as mentioned above, is fully transitive.

A. Moskalenko [10] proved that when T is a direct sum of cyclic p -groups, • then T is also fully transitive and all the conditions of Theorem 1 are fulfilled.

Therefore in this case, too, the *Ω of filters of indicators describes the lattice of fully invariant subgroups.
The direct sum of torsion- lattice complete groups is a natural generalization of direct sums of cyclic p -groups and torsion-complete groups.

As the author has shown in [13], in this class of groups the cotorsion hull is not fully transitive if the sum is infinite.

Hence, because of condition 4) of Theorem 1 the lattice of fully invariant subgroups cannot be described by means of indicators.

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1, 2010 The Lattice of Fully Invariant Submodules of a Reduced Cotorsion p-Adic Module 19 In [14] the lattice of fully invariant subgroups of the group • T is studied in the case, in which T is a countable direct sum of torsion complete p -groups: where jB is a basic subgroup of ,jB and ∞ B ⊕ = = 1 j B j is a basic subgroup of T .
When T is a separable p -group, T ∞ ⊕= = j 1 B j , (2) T are represented as countable sequences (see [10]) elements of the cotorsion hull { ( = ∈ = , aa o 1 ∈− a i ,... aT ) ,... , iT ,ˆ T pa + 1 T T + + a • i i i • }.

...1,0 Using this representation of elements, we can easily calculate their height and indicator.

, Let be some fixed basic subgroup of the separable p -group T .
If a ∈ • T B x ⊕= ∈ Ja a = a ( aa , 0 1 + T ,...), then the group B contains a sequence ( bi ), =i ,...,1,0 such that for any i = b i s ∑ = 1 j xm jaj , ≤ 0 m j < p , and = a i ⎛ ⎜ lim ⎜ ∞→ n ⎝ n ∑ = 0 s s bp + si ⎞ ⎟ .

⎟ ⎠ (3) This representation of an element a is called canonical.

We say that the sequence ( ib corresponds to the ) canonical representation of an element a .

Let a group T have form (2) and a •∈T (see (3).


Denote by π the projection of the group T on the direct i summand iB and consider the sequence π i b ( ) j = b ( ji ), = j ,....1,0 For every 1≥i and fixed j, the sequence b i 0 , b i 1 ,... , defines the element = a ji lim ∞→ n n ∑ = 0 s s bp + sji , while the elements , a i a 0 i 1 ,... of the group iBˆ define the element a i )( = ( a i 0 , a i 1 + T ,...) of the group • T .

It is obvious that = a j lim ∞→ n n ∑ = 1 i a ji .
Note that the elements ,)( a i =i ,2,1 ... sequence corresponding to the element a and π = (ˆ i ∈′− a k π (ˆ, k , then ' ) k if T = a a a a ) k k induced projection on the group ki .ˆ iB are uniquely defined by the element a from (3) since if ( ' jb ′ +′ , aa 0 1 =′− π (ˆ a ik i , ( aa 1 0 ' and ki ,...), T ∈′− ) k ′= a 0 ,...2,1 then kT , a = ,...) T + = a a a a ik ( 0 k (see [10, §§1, 2]), and πˆ denotes the , where i ) j is another ≥ 0 To each element •∈ Ta we assign the matrix where aH ( = ) 00 aHaH ), ( ( • T = ) i 0 Definition 1.
The matrix ijk , ≥ji 0 i 0 ) for aH ( ˆ T made up of ordinal numbers and symbols ∞ is called admissible with respect aH i ( 0 ,) ≥i ,0 Φ )(a = .1≥i to the group T if the following conditions are fulfilled: 1.

The first row is an increasing sequence of ordinal numbers smaller than ωω+ or of symbols ∞ , and if ω≥ , The other rows are increasing sequences of non-negative integers or , then n ≥ + k =+ .j jk k 0 1 symbols ∞ (it is assumed that ∞ + 1 = ∞ ).

m for any 2.
m for any 2.

If 1 on n 0 + kon = ω in the first row is the first infinite ordinal number and for contain a non-negative integer and there exists a row 0i such that starting from some i0, all rows consist only of symbols ∞.

∞=−mink ,nm < i ≥ .0i then infinitely many rows then, , nmm = ω + ≥ , If k n 0 3.

If all elements in a row are non-negative integers, then this row contains infinitely many jumps.

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1, 2010 20 Tariel Kemoklidze 4.


If between B = 0 assumed that ijk j and ).

B 1+ijk there is a jump, then in the group iB there exists a base element of order 1+ ijkp (it is ∞→ as ∞→i ; also, if k j 0 ≠ ω then m , + k = oj k min{ , k 1 j 2 j ,...}, and if k j 0 += ω m , j 2 k ... k 1 We denote the set of admissible matrices (with respect to T ) by Ω and consider in it the reflexive and transitive j 5.

In every column ∞= = = ijk .

then relation ≤ (see [13], Definition 1.2).


Then the relation [ ≤⇔ U ρ VU V and ]UV ≤ is the equivalence on the set Ω , and the relation ≤ VU ≤⇔≤ VU defined on the quotient set Ω=Ω ρ/ is the order.

Φ • : T Φ=ΦΩ→ ( ) a ( )a • In [14] it is shown that the function T , where T has form (2) and Ω is the set of admissible matrices with respect to T , satisfies all the conditions of Theorem 1.

Therefore the following statement is true.

defined on the group , Theorem 2.


The lattice of fully invariant subgroups of the cotorsion hull • T , where T is a countable direct sum of torsion-complete p -groups, is isomorphic to the lattice of filters of the semilattice Ω .

Let us return to the reduced cotorsion p -adic module (1), where T has form (2), and investigate the lattice of fully invariant submodules of the module A .
To this end, we consider the set Ω=Ω • HU , where Ω is the semilattice figuring in the formulation of the Theorem 2 and H is the set of increasing sequences of there is a jump, then in the nonnegative integers, containing only a finite number of jumps.

If between 1+ basic subgroup B of the group T there exists a base element of order ik and 1+ik ikp .

define the relation ≤ on the set the set Ω .


Otherwise, it is required that the following condition should be fulfilled: The elements of the set Ω are the classes of admissible matrices, where the first row is uniquely defined.

Let us , then K ≤ 'K is defined in the same manner as in the case of where are the first rows of matrix taken from the classes of the set Ω or sequences from H .

It is obvious •Ω .
It is obvious •Ω .

If K and ,...1,0=i Ω∈'K k ≤ i ≥ k ' i i ′ i i k 0 ), (), ( k that the relation ≤ is a relation of order on the set , then the exact lower bound inf ( If •Ω .

, KK )' , KK Ω∈' done for the set of Ω .

If ) ( ) ( min , where ( k j k 0 , ′ j ≥ j 0 Ω∈K 0 ) jk ≥j 0 of these elements is defined in the same way as this was =) and is the first row of an admissible matrix K .
Taking into account the definition of a are assumed to be given, then we have inf( , HK ...) ' k 1 H H ∈ = ' o k ( , , matrix K , we obtain inf ( , HK ) ∈ Analogously, we define H .

inf( 1 HH , 2 ) for that this definition satisfies the requirements for an exact lower bound.

Hence the set 2 H∈ .

It is easy to verify 1, HH •Ω is the lower semilattice.

Let us consider the map which to each element •∈Ta assigns )(aΦ or, if • • Ω→Φ A: += t c a , where c ≠ ,0 ∈ Cc is the p-indicator of an element a .
It is not difficult to verify that the function Theorem 1.

Φ• aHa = )( ( , then )(aH •Φ satisfies all the conditions of where ), Thus the following statement is valid.

Theorem 3.

The lattice of fully invariant submodules of a reduced cotorsion p -adic module A is isomorphic to the lattice of filters of a semilattice •Ω .

Acknowledgement.

This study was supported by grant ATSU/09316.

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1, 2010 The Lattice of Fully Invariant Submodules of a Reduced Cotorsion p-Adic Module 21 maTematika reducirebuli kogrexviTi p-adiuri modulis savsebiT invariantul qvemodulTa meseri t. qemokliZe a. wereTlis saxelmwifo universiteti, quTaisi (warmodgenilia akademikos x. inasariZis mier) naSromSi ganixileba reducirebuli kogrexviTi p-adiuri modulis savsebiT invariantul T ⊕ C saxe, sadac T aris grexviTad sruli p-jgufebis qvemodulTa meseri, roca moduls aqvs Tvladi pirdapiri jami, xolo C _ grexvisagan Tavisufali algebrulad kompaqturi jgufi.
naCvenebia, rom aRniSnuli meseri usasrulo matricebisa da indikatorebisagan agebuli naxevarmesris filtrTa mesris izomorfulia.

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Received October, 2009 Bull.

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1, 2010