ABSTRACT

The paper considers the problem of full transitivity of a cotorsion hull of a separable primary group when a ring of endomorphisms of the group has the form, where is a subring of small endomorphisms of the ring, whereas is a ring of integer -adic numbers. Investigation of the issue of full transitivity of a group is essentially helpful in studying its fully invariant subgroups as well as the lattice formed by these subgroups. It is proved that in the considered case, the cotorsion hull is not fully transitive. A lemma is proposed, which can be used in the study of full transitivity of a group and in other cases.

Keywords:Full Transitivity of a Group; Cotorsion Hull; Fully Invariant Subgroup

1. Introduction

The groups discussed in the paper are abelian and the operation is written in additive terms. We use here the notation and terminology of the monographs [1] [2] .

The symbol denotes a fixed prime number. and are respectively the groups of integer and ratio­nal numbers. A subgroup of the group is called fully invariant if it is self-mapped for any endomorphism of the group.

The knowledge of the construction of fully invariant subgroups of an abelian group and their lattice is essentially helpful in the study of the properties of the group itself and also in the investigation of the properties of its rings of endomorphisms and quasi-endomorphisms, the group of automorphisms and other algebraic systems connected with the initial group.

For a sufficiently wide class of -groups these topics were studied by R. Baer, I. Kaplansky, P. Linton, R. Pierce, D. Moore, E. Hewett and others. The works of A. Mader, R. Göbel, P. A. Krylov, S. Ya. Grinshpon, A. I. Moskalenko and other authors are dedicated to the investigation of these topics in torsion-free and mixed groups.

However little is known about the results obtained in this area for the class of cotorsion groups. A group is called a cotorsion group if its extension by means of any torsion-free group splits as follows:

. The importance of the class of cotorsion groups in the theory of abelian groups is due to two factors: for any groups, , the group is a cotorsion one and any reduced group is isomorphically embeddable in the group called the cotorsion hull of the group. If the torsion part of the group is denoted by, then

where. Thus the study of cotorsion groups essentially reduces to the study of groups of the form, where is a -primary group.

It is noteworthy that endomorpohisms in cotorsion groups are completely defined by their action on the torsion part and, as shown by W. May and E. Toubassi [3] , for a mixed group the ring of endomorphisms is isomorphic to if and only if is a fully invariant subgroup of the cotorsion hull.

The notion of full transitivity of a group plays an essential role in describing the lattice of fully invariant subgroups.

By the -indicator of an element of the group we mean an increasing sequence of ordinal numbers

where is the generalized -height of an element, i.e. for if and. Now for the set of indicators we can introduce the order

A reduced -group is called fully transitive if for arbitrary elements and, when there exists an endomorphism of the group such that. The class of fully transitive groups includes such important groups as separable -groups, algebraically compact groups and quasi-pure injective groups.

Using the indicators of fully transitive groups we can describe the lattice of fully invariant subgroups (see [4] -[11] ).

For a module over a commutative ring, A. Mader formulated a general scheme that can be used to describe the lattice of fully invariant submodules of the module (see [10] , Theorem 2.1 or [12] , Theorem 1.1).

In the same way as we did for a -group we define the notion of full transitivity for the group

. According to A. Mader [10] , an algebraically compact group is fully transitive and described with the aid of indicators the lattice of fully invariant subgroups of this group. This means to describe the lattice of fully invariant subgroups of the group when is a torsion-complete group. When is the direct sum of cyclic -groups, A. Moskalenko [11] proved that is also fully transitive and described by means of indicators the lattice of fully invariant subgroups of the group. In general, for the separable primary group, the cotorsion hull is not fully transitive. In particular if is an infinite direct sum of torsion-complete groups, then, as shown by the author [13] , the group is not fully transitive and in that case the lattice of fully invariant subgroups of the group cannot be described by means of indicators (see [12] ).

R. Pierce [14] considered the primary group, a ring of whose endomorphisms has the form

(1.1)

where is the ring of small endomorphisms of the group which is the ideal of the ring of endomorphisms of the group, whereas is the ring of integer -adic numbers. A small endomorphism of the group is defined as follows (see [14] ).

For all there exists an integer such that

. (1.2)

The Pierce group is important when studying the ring of endomorphisms of abelian groups (see [15] ). The aim of the present paper consists in elucidating the full transitivity of the cotorsion hull and also in finding the conditions, under which the cotorsion hull is not fully transitive.

2. Full Transitivity of the Cotorsion Hull of the Pierce Group

As mentioned above, R. Pierce [14] considered the separable primary group with a standard basic subgroup

, , , , where is a torsion-complete group, i.e. the torsion part of a -adic completion of the group. The cardinality is and the ring of endomorphisms of the group has form (1.1).

To study the full transitivity of the group, we use the following representation of elements of the cotorsion hull of given by A. Moskalenko [11] for the separable -group

. (2.1)

Representation of elements in this form makes it easy to calculate the height and the indicator. In particular, if, then

(2.2)

where is the smallest infinite ordinal number.

Let be a basic subgroup of the reduced separable -group lying between and. Elements, ,. As is know, an endomorphism of the group extends uniquely to an endomorphism of.

The following lemma is true.

Lemma 2.1. If and there exists no endomorphism of the group for which, then a cotorsion hull is not fully transitive.

Proof. Consider two elements

of the group. Then by the condition of the theorem and (2.2) we have. As is known, each endomorphism of the group extends uniquely to an endomorphism of the group. We will show that if for an endomorphism, , then. Let

(2.3)

be the element of the group defined by the sequence. For an endomorphism of the group let us show that. According to ([1] , Section 50), the extension of is defined from the commutative diagram

(2.4)

where is the identical inclusion,

The commutativity of diagram (2.4) immediately follows from the definition of these homomorphisms.

To extension (2.3) there corresponds the sequence, where elements are defined as follows: fix a system of generators of the group, ,. Let, , be a system of representatives of the adjacent classes of the group, , ,. Denote

.

Then for each,

.

For an endomorphism of the group we have, , and can define

(2.5)

It is obvious that the right-hand part of equality (2.5) defines the extension of an endomorphism on and if is some other endomorphism of the group, which induces on, then contains and ([1] , Proposition 34.1). From (2.5) we have

.

Now we can consider an element

(2.6)

of the group and with its aid define the corresponding short exact sequence.

Let be the group defined by a system of generators which are defined by the relations of the group and the equalities, ,. Then

(2.7)

where is the identical inclusion and, for each element, , ,  is a short exact sequence. To extension (2.7) there corresponds sequence (2.6) (see [11] , Proof of Theorem 1). Let us show that by using extensions (2.3) and (2.7) we can compose the commutative diagram

(2.8)

where is the above-mentioned endomorphism and, , ,. Indeed, from the definition of a triple we immediately conclude that (2.8) is a commutative diagram.

Thus we have shown that (2.4) and (2.8) are commutative diagrams. Then, according to ([1] , Section 50), and are equivalent extensions and thereby define one and the same sequence from. But, by virtue of our construction, is the sequence corresponding to the extension; therefore it corresponds to the extension, too. Thus. Therefore if the endomorphism maps the element into, then, i.e. we have proved more than what has been mentioned at the beginning of the proof of the lemma. Thus it obviously follows that if there exists no endomorphism of the group for which, then there exists no endomorphism of the group which maps the element into, i.e. is not fully transitive. The lemma is proved.

For the Pierce group the following statement is true.

Theorem 2.1. The cotorsion hull of the group is not fully transitive.

Proof. We use representation (2.1) of cotorsion hull elements and assume that and are elements of the group, where,. By virtue of (11, Item 2), elements and can be written in the form

where,. Since and is infinite, taking into account ([14] : Lemma 15.1, Theorem 15.4) we can assume that

(2.9)

By (2.2) we have, i.e. the following condition is fulfilled

.

Let be an endomorphism of the group. Using (1.1) we have, where is a small endomorphism of the group and is the -adic number. As is known ([1] , Section 39), the endomorphism uniquely extends to the endomorphism of the group

Since

(2.10)

and is a small endomorphism of the group (see (1.2)), starting with some we have

.

Therefore

On the other hand, from (2.10) we obtain

Therefore

But since, , and in that case the equality would contradict condition (2.9). Therefore

.

Thus there exists no endomorphism of the group which extends to the endomorphism of the group and. Then from Lemma 1.1 it follows that Theorem 2.1 is valid.

Note that one more example of a separable primary group, the cotorsion hull of which is not fully transitive, can be found in ([11] , item 3).

As mentioned above, if the separable primary group is a direct sum of cyclic -groups or a cotorsioncomplete group, then the cotorsion hull is fully transitive. In 1993, at Professor A. Fomin’s seminar A. Moskalenko made a conjecture that is fully transitive only in these two cases. The proved lemma and theorem may serve as a positive argument in favor of this conjecture.

Acknowledgements

This study was supported by the grant (ATSU-2013/44) of Akaki Tsereteli University.

References