Near-isomorphism is known as the right concept for classification theorems in the theory of almost completely decomposable groups. As a natural generalization the authors extended in  the notion of near-isomorphism to Abelian groups of arbitrary rank. In this article we investigate non-isomorphic direct decompositions of a class of infinite rank torsion-free abelian groups which were defined in  as special epimorphic images of so-called almost rigid groups. A complete classification of such decompositions up to near-isomorphism is given.
In the theory of Abelian groups we know many examples where a classification of groups up to some reasonable variant of isomorphism has successfully been obtained. In particular, a characterization by numerical invariants has been of interest. For instance, completely decomposable groups (direct sums of subgroups of the rationals) were classified up to group isomorphism by so-called types and ranks of their homogeneous components. In the torsion case and later on in the mixed case Ulm’s invariants and Warfield’s invariants were used to classify totally projective Abelian -groups and Warfield groups up to isomorphism. A very good source on these results are the two volumes by Fuchs  and Loth . Focusing on the class of torsion-free Abelian groups however, it is in general very difficult (and very often impossible) to obtain such a classification. A very good example where a modified notion of isomorphism led to success in this respect is the class of almost completely decomposable groups (acd-groups) which are finite extensions of completely decomposable groups of finite rank (see Mader ). It turned out that for acd-groups the concept of near-isomorphism was more suitable than classical isomorphism. Recall that two torsion-free abelian groups X and Y of finite rank are nearly-isomorphic, in symbols , if and only if for every prime p there is a monomorphism such that is finite with zero p-torsion part. Near-isomorphism is indeed an equivalence relation (so implies ) and from many perspectives almost completely decomposable groups under near-isomorphism are very attractive, in particular, concerning their direct decompositions and the links with their endomorphism rings: nearly-isomorphic acd-groups have isomorphic regulators, nearly isomorphic endomorphism rings (considered as Abelian groups), and sometimes even allow a sort of Baer–Kaplansky theorem. Finally, their direct decompositions in the case of so-called crq-groups (i.e. finite cyclic extensions of completely decomposable groups) can be classified up to near-isomorphism in terms of the so-called type-invariants (see the standard book by Mader  on acd-groups and the references in there).
There are several ways to extend the class of almost completely decomposable groups to infinite rank, e.g., by looking at continuous increasing chains of acd-groups or by considering torsion-free Abelian groups of infinite rank with all their finite rank pure fully invariant subgroups isomorphic to almost completely decomposable groups – called local acd-groups. In each case it is also desirable to have an appropriate notion of near-isomorphism. Such concept was introduced by the authors in [6, Definition 2.3, Proposition 3.4] by defining two torsion-free Abelian groups X and Y of arbitrary rank to be nearly-isomorphic ( ) if for every prime p there exist monomorphisms and such that
and are torsion,
for all finite rank pure subgroups and the quotients and are finite.
It was shown in  that this notion extends the classical notion of near-isomorphism for finite rank groups (forced by condition (3) above) and for completely decomposable groups it is equivalent to isomorphism as one should expect. Thus keeping the name near-isomorphism is justified.
Our goal in this paper is to classify up to near-isomorphism all possible direct decompositions of countable rank torsion-free Abelian groups from a special class defined in . The first results in this direction were obtained in  and later on in  for a class of Abelian groups where a group X in is generated by a completely decomposable subgroup of X – the so-called regulator – with critical typeset an antichain and τ-homogeneous completely decomposable pure subgroups of such that the quotient is a direct sum of bounded p-groups. Moreover, the relations defining X with respect to are of a finite character (see  for further details). A complete classification up to near-isomorphism was obtained in the series of papers [1, 3, 6] for different subclasses of assuming that the ranks of the homogeneous components of are finite (this class was denoted by in ): the class of rigid groups, i.e. groups such that the rank of any homogeneous component is one, and the class of almost rigid groups, i.e. groups with homogeneous components of finite rank, the critical typeset is countable and all primary components of are cyclic groups. In fact, almost rigid groups have a decomposition into a direct sum of a completely decomposable group and a rigid summand from which explains the name almost rigid. In particular, it was shown in  and [1, Theorem 3.4] that for countable rank groups from the class , in particular for almost rigid groups, a classical theorem by Arnold can be extended: If X and Y are nearly-isomorphic and , then there is also a decomposition such that and (for the reader’s convenience we should say that the class of countable rank groups from was denoted by in the paper  published in Russian). Moreover, rigid groups and also almost rigid groups from were classified up to near-isomorphism by their regulators and certain numerical invariants (see  and ) and a decomposition theory was constructed in  in terms of the numerical invariants as a development of the combinatorial approach invented for some almost completely decomposable groups in  and [10, Chapter 13]. Finally, the authors and Göbel passed to certain epimorphic images of rigid groups and also almost rigid groups, called proper asd-groups and proper alr-groups, which are obtained by factoring out a subgroup K of an (almost) rigid group X. The subgroup K is of the form , where and the regulator of X satisfies
(for a more detailed definition see Definition 2.2 below). The integers , and the sets of types are called a triple of partitions, coefficients, and parameters for X. For proper asd-groups and proper alr-groups a classification up to near-isomorphism is available (see ) in the sense that two such groups and are nearly-isomorphic if and only if their pre-images X and Y are nearly-isomorphic and the triples of partitions, coefficients, and parameters for X and Y coincide.
In this paper we extend the numerical approach for (almost) rigid groups to the proper alr-groups using the near-isomorphism invariants from : Two proper alr-groups B and C are nearly-isomorphic if and only if and for each critical type (Theorem 3.3). Moreover, two classification theorems up to near-isomorphism of direct decompositions of proper alr-groups are obtained by carrying over the near-isomorphism classifications for almost rigid groups to the class of proper alr-groups (see Lemma 4.4 and Theorem 4.6).
The main result of the present paper is the direct decomposition classification in Theorem 4.6 in terms of the numerical invariants for proper alr-groups. These groups have many common features with almost rigid groups. However, the big difference is that they are torsion extensions of direct sums of strongly indecomposable groups while almost rigid groups are torsion extensions of completely decomposable groups.
2 Preliminaries and notation
All definitions and group properties, which form the basis of our considerations, can be found in the two standard books by Fuchs  and Mader . The latter contains special notions of the theory of almost completely decomposable groups which we will use and extend in this paper to some classes of infinite rank groups X of ring type, i.e. groups having critical typesets consisting only of idempotent types τ. Such idempotent types can be represented by characteristics with only s and s as entries, see [10, p. 13], [7, Section 85]). We write if τ has as entry at position p and for the other p. Then is a τ-homogeneous group, containing the element a, that is divisible only by primes p with .
Moreover, the groups X under consideration are block-rigid (i.e. with antichains as their critical typesets) and, therefore, for any the fully invariant pure subgroup is a homogeneous group of type τ denoted by , and which will be called a τ-homogeneous component of X. As for other notation, we use to indicate that Y is a pure subgroup of the torsion-free group X, and for the p-component in the primary decomposition of a torsion group T into primary summands. In most cases we say “group” for a torsion-free abelian group X because it is our main object and denote the subgroup Y of such that by with natural q.
A torsion-free abelian group X belongs to the class if there exists a completely decomposable subgroup of X such that the following conditions are satisfied:
is an antichain of idempotent types,
is a finite rank pure and τ-homogeneous completely decomposable subgroup of X for all ,
for some set of primes and -bounded p-groups ,
for every the set is finite; here, is the minimal subset such that
We will be interested in factor groups of almost rigid groups, a subclass of the class . Recall from  that a group is called an almost rigid group (alr-group) if the set is countable and all primary components of are cyclic groups. Knowing their classification from  we will use the almost rigid groups as preimages of the groups, which are to be classified here up to near-isomorphism. The following recalls [4, Definitions 5.1, 5.3]:
Let be an almost rigid group and let be integers such that the following conditions hold:
(where ) with , the countable disjoint union of finite subsets and for any ,
for any and any there exists a prime p such that and for all , ,
if and , then for any ,
if and , then is not p-divisible if for some ; if , then with ,
if and , then for any ,
for any and ,
if and for some , then each is relatively prime to p for any ,
if and , then for any .
Let be defined as follows:
The partition and the two sets of integers , will be called a triple of partition, coefficients and parameters for the group X. Moreover, the group is called a proper alr-group and the canonical epimorphism will be called a regular representation of the proper alr-group B. The regulator of B is defined as .
Note that for simplicity we use to denote the direct sum of copies of in the above Definition 2.2.
Recall that for an almost rigid group there are elements such that
for some integers since the primary components of are assumed to be cyclic. Writing
where is the canonical epimorphism. The numerical invariants of X were then defined as
The appearance of these numbers for proper alr-groups will be discussed in detail below, but it was already shown in [4, Definition-Lemma 5.6] that conditions (6) to (8) of Definition 2.2 can be reformulated using the numerical invariants :
Conditions (6)–(8) of Definition 2.2 are equivalent to the following ones:
if there exists with and , then ,
if there exists with and , then is relatively prime to ,
if there exists i with and , then for any prime divisor p of .
The basis of our desired decomposability criterion for proper alr-groups will be the corresponding decomposability criterion [3, Theorem 5.2] for almost rigid groups.
Theorem 2.4 ([3, Theorem 5.2])
Let X be an almost rigid group with and . Then there exists a decomposition into almost rigid summands with products , where if and only if the following compatibility conditions hold:
The numbers and are relatively prime if and .
for any .
In order to carry over this decomposition result for almost rigid groups to their images, the proper alr-groups, we will need the so-called admissible decompositions for almost rigid groups (see [4, Definitions 5.8, 5.9] ):
Let be an almost rigid group that has a decomposition . If ( ) are rigid groups, having regulators isomorphic to direct sums of groups from the set , then this decomposition will be called an admissible decomposition of X.
Among all the admissible decompositions of almost rigid groups there are special ones, determined uniquely up to near-isomorphism and called main decompositions:
Let be an almost rigid group. An admissible decomposition , where is a rigid group satisfying:
if and only if for some ,
will be called a main admissible decomposition of X.
It will be shown next that we are interested only in admissible decompositions of an almost rigid group X. In fact, given X and a proper alr-group that has a decomposition, then our strategy is to see that there is an admissible decomposition of X such that the direct summands can be considered as preimages of the summands of B. More precisely, we need some restrictions on the admissible decompositions of X which will lead us to the notion of K-admissible decompositions in the later sections.
3 Near-isomorphism of alr-groups in terms of the invariants
Our aim in this section is to introduce numerical invariants for proper alr-groups and to obtain a near-isomorphism criterion for these groups formulated in terms of these invariants. This will be done in complete analogy to [4, Theorem 5.15], which is the main result of that paper.
Let be an almost rigid group and a proper alr-group with the regular presentation of B. For simplicity we denote in (3) of Definition 2.1. It is almost evident and easy to recall from [4, equations (2), (3), and (12)] that there exists a uniquely determined factor primary representation of X in the following form:
This implies the factor primary representation of B:
Besides with , we introduce the set of finite rank fully invariant subgroups of X
Their images are isomorphic to and they are block-rigid crq-groups with primary regulator quotient. It follows that there exist elements such that with and, clearly,
with if .
Recall from [4, Theorem 5.13] the following
Let be a regular representation of a alr-group B with the triple , and of partition, coefficients and parameters for the group X. Then there exists a main admissible decomposition with and the corresponding decomposition such that , .
In [4, Definition 5.14] this fact motivated a new definition as follows,
Given a proper alr-group B, a decomposition with a rigid alr-group such that will be called a main decomposition of B.
Now, if is the main decomposition of the proper alr-group B, induced by a main admissible decomposition , which means that and , then let
be the factor primary representation of the rigid group , see (3.1).
with finite rank rigid crq-subgroup
of for each .
Let with . Define in uniquely determined crq-subgroups , then , .
We may define the set of numbers
with the property , . Furthermore, and we are allowed to put
Recall again from [3, Theorem 4.12] that two almost rigid groups X and Y are nearly-isomorphic if and only if their regulators are isomorphic, in symbols , and for each critical type and using the fact that by [4, Theorem 5.15] two proper alr-groups are nearly-isomorphic if and only if their preimages in regular representations are nearly-isomorphic and the triples of partition, coefficients and parameters coincide for them we immediately obtain the following near-isomorphism criterion:
Let B and C be proper alr-groups. Then if and only if and for each critical type .
Having this result we are permitted to call the numbers not only type-invariants of a proper alr-group B, but also near-isomorphism invariants of B, . The same terminology is appropriate for almost rigid groups, which are preimages of alr-groups.
4 Direct decompositions of alr-groups
We are now approaching our main result which gives a combinatorial decomposability criterion for proper alr-groups based on that for almost rigid groups, see Theorem 2.4. In connection with Definition 2.2 we introduce some more definitions. Recall that a proper alr-group was defined as the quotient of an almost rigid group X and some subgroup K of the following form:
where with .
Any set of elements will be called a K-basis of if the following hold:
Automorphisms will be called K-basic automorphisms of X if for any they act on each τ-homogeneous fully invariant subgroup as multiplication by the same matrix ( ) with respect to a K-basis .
As was promised above we strengthen the definition of admissible decompositions as follows, see Definition 2.5.
Let B be a proper alr-group and let be its regular representation. We say that is a K-admissible decomposition of X if it is admissible and obtained by a K-basic automorphism of X.
At this point we would like to remark that the K-admissibility of a decomposition of the almost rigid group X is equivalent to the existence of the corresponding induced decomposition with (see Definitions 4.1 and 4.2). It then follows that the proper alr-group is also decomposable as . This is the reason why we needed the K-admissible automorphisms and this leads to the following very natural
Lemma 4.4 (Decomposability Criterion 1)
Let B be a proper alr-group and its regular representation with the triple , and of partition, coefficients and parameters for the group X. Then one has if and only if there exists a K-admissible decomposition such that for .
The sufficiency of the condition is trivial by the remark above.
Let . This implies the existence of the corresponding decomposition of its regulator, fully invariant subgroup with and . Then we have and and are also alr-groups. We have which is isomorphic to . Then there exists the partition such that and .
For the same reason we have the corresponding decompositions of the groups with critical typesets , which are direct sums of copies of strongly indecomposable groups , see [7, Section 92] and [8, Section 3.3]. Namely, with and for each since are fully invariant in , see [3, Introduction]. Denote and let be its fully invariant τ-homogeneous subgroup with . Then with one of the summands allowed to be zero. Recall that , which implies the corresponding decomposition of the regulator of X with and .
Clearly, , and is a K-admissible direct decomposition such that , and , . ∎
Since the number of direct summands of B is at most countable (because it is not greater than the rank of the group), a trivial induction leads to the following.
Let B be a proper alr-group and let be its regular representation with the triple , and of partition, coefficients and parameters for the group X. Then if and only if there exists a K-admissible decomposition such that , .
Now we need some preparation for transforming the obtained decomposability criterion into the one based on the numerical invariants of alr-groups which will be our main result.
Let M denote the set enumerating the groups to write for our convenience. Then , see Definition 2.2. For any filtration with and there exists a corresponding filtration with the ascending chain of fully invariant pure subgroups
such that and . Note that all are crq-groups. Without loss of generality assume that it is a special filtration characterized by the condition that and are relatively prime for any and .
For any introduce a K-basis of as , see Definition 4.1.
Denote , . Recall from [3, Lemma 4.6] that
if . Moreover, near-isomorphism invariants of the groups can be calculated on the basis of [3, Proposition 2.11], namely,
with , .
If , then
as is fully invariant in X. Moreover, near-isomorphism invariants of and can also be calculated in the following way:
if we naturally denote
In the proof of the next theorem we need to construct decompositions of the proper alr-group with type invariants induced by the decomposition of the almost rigid group . So, we will say that the invariants and are predicted (or even a decomposition of is predicted in the sense that its regulator and near-isomorphism invariants are known).
The next ingredient of the algorithm will be the extension of a decomposition (4.2) from to , which means that and .
Following  we say that a direct sum of strongly indecomposable groups is a strongly decomposable group. It is associated with completely decomposable groups which are direct sums of rank-one groups.
Finally, we have approached the main result summarizing the above discussion. Let
be a special filtration of X with an ascending chain of fully invariant pure subgroups satisfying the following condition: and are relatively prime for any and .
The following main Theorem 4.6 is a generalization of the corresponding Theorem 5.2 from  on almost rigid groups to the class of proper alr-groups. Its proof is quite technical because it needs a double induction. First, we restrict ourselves to the case of two summands and induct on . Then we induct on the number of the summands.
Theorem 4.6 (Decomposability Criterion 2)
Let B be a proper alr-group with . Then there exists a decomposition
into rigid summands with and strongly decomposable C if and only if the following hold:
for any ,
, , .
Let be the regular representation of B. On the basis of Decomposability Criterion 1 (Lemma 4.4) we have that the decomposition
with satisfying (1)–(3) exists if and only if there is a K-admissible decomposition
with , and , see (3.2).
Note that the required admissible decomposition (4.3) of X with conditions (1)–(3) is one of the decompositions described in the decomposability criteria [3, Theorem 5.2]. Then we need only to show that it can be obtained by a K-basic automorphism of X. Moreover, the necessity of these conditions also follows from the mentioned criteria.
To prove that conditions (1)–(3) are sufficient for the existence of the required decomposition of B we take its main decomposition , which exists by Theorem 3.1, and obtain a decomposition (4.3) on this basis. Without loss of generality assume that for any i with . Since there exists a corresponding admissible main decomposition
with and we need to obtain from (4.4) by a K-basic automorphism an admissible decomposition
into rigid alr-groups of rank more than one and . We may restrict ourselves to the situation when for each i satisfying with .
More precisely, our purpose is to get the corresponding decomposition
We may also assume that . If , then H contains a completely decomposable summand , therefore and can be removed from (4.4) and (4.5) accordingly and in this case for any there exists such that . Moreover, without loss of generality assume that there is no non-trivial partition such that whenever and . Otherwise group B is a direct sum of proper alr-groups with independent direct decomposition constructions. It follows from (3.2) that the same connection condition should be assumed for group X, that is for any partition there exist and such that and are not relatively prime.
So, we need to have the necessary decompositions of alr-groups.
Case I. Let us first consider the case , that is and for all . By Lemma 4.4 we need a decomposition with not necessarily indecomposable rigid summands and invariants , , . For the corresponding decomposition of the regulator such that and we have that for each the following holds:
with and (remark that or is allowed to be zero). We are concentrated on the admissible decompositions of X, therefore, if for some then for any with , .
We now intend to obtain by a K-basic automorphism such a decomposition
from the main decomposition (4.4) with , .
Clearly there exists a K-basis of such that
with if , see (4.4).
Let be an arbitrary fully invariant pure subgroup of X with . We are now concentrated on with the regulator . Recall that .
Since is fully invariant in X, the above decomposition , which is to be constructed, reflects on the group in the way that it must have the corresponding decomposition with , . Similarly, with , etc.
Recall from [3, Lemma 4.6] that we have if , and if .
Our first object now is to get the necessary decomposition of . To this end, Let , and . These numbers can be viewed as , and . We also have that
with , , and also
with , . Evidently, is equal to 1, see conditions (1)–(2) for .
By number theory there exist relatively prime integers and such that , that is
Moreover, we can find integers u and v such that .
Determine a new K-basis of as follows:
For each the matrix of linear transformation of is
If for some i, that is for all , then coincides with or and we take accordingly
see (4.8). Therefore
Denote and . By construction,
then and . Hence the K-basic automorphism Ψ of X, which acts on each τ-homogeneous component of rank 2 as multiplication by the matrix D with respect to and is identity, if , led us to the predicted K-admissible decomposition of determined by (4.6).
The next purpose is to get the admissible decomposition of determined by (4.6) which preserves the decomposition of in the following sense:
Let α and β be some integers such that
Put and we have and also
It is important that the two numbers and are relatively prime and as well as by construction as and . Then the numbers and are relatively prime and there exist integers , such that the matrix
Now we are able to construct the required decomposition of . Determine a K-basis of as follows:
for each .
If for some i, that is for , then coincides with or and we take accordingly
Denote and .
Since the elements of the first lines of the matrices D and are congruent modulo and accordingly by (4.11) as and , we obtain that and , which means
Hence the K-basic automorphism Ψ of , which acts on each τ-homogeneous component of rank 2 as multiplication by the matrix D or depending on or with respect to and it is identity, if , led us to the required admissible decomposition of determined by (4.6).
Continuing the process we take the groups , and and consider them as , and respectively to apply for them the above construction. Instead of D we take the constructed matrix , the numbers and will serve as and , the numbers and will serve as v and u and the numbers and will work as the new and accordingly. Due to the above connection restriction we get the required decomposition of for any in finitely many steps.
Case II. Consider the general case , which implies for each that can be any finite positive integer. Under the condition we have to construct a decomposition
into rigid summands of rank more than one with a set of near-isomorphism invariants satisfying conditions (1)–(3) on the basis of the main K-admissible decomposition (4.4). Without loss of generality assume that the rigid summands are listed in (4.12) in the following order: are all the summands having non-zero intersection with , are all the summands having non-zero intersection with and so on. Assume that (we are able to choose satisfying this condition, otherwise the decomposition has been already constructed in Case I). The case is allowed for some (or even all) .
As above, is a K-basis of . It follows from the decomposition in (4.12) that if because, by construction, . Let be the minimal natural number such that (equivalently, is the natural number satisfying or, the same, ). Then we may rewrite the K-basis of each , , in the following way:
First we should concentrate on , and again as in Case I. The main decomposition of determined by the main admissible decomposition (4.4) is
Let us take a subgroup of with homogeneous components of rank not greater than 2, which is
and construct its K-admissible decomposition with the rigid summand , uniquely determined up to near-isomorphism by its near-isomorphism invariants
under the restriction in (4.13) that if , see Case I.
The connection condition provides that a special numbering the homogeneous components , , yields that and are not relatively prime for some , . Again using Case I we extend the obtained decomposition of to , which is
with and pure fully invariant subgroup . Then it follows that with the rigid summand having the predicted invariants, and rigid summand , uniquely determined up to near-isomorphism by its regulator and the numbers
It is important that and (in particular, the possible equalities or do not create any problems), see (4.10).
Since , we have a subgroup of which is
and construct its K-admissible decomposition with the predicted and rigid summand satisfying
Extending this decomposition to the group
we obtain with predicted and
satisfying and (as above, the case is obviously allowed).
Continuing this process in steps we obtain a decomposition
with the predicted invariants of rigid groups with , rigid satisfying
and completely decomposable G with . Note that by construction
Moreover, for each and . If , then . If , then the summands of with have zero intersection with and can be obtained on the basis of the main decomposition as above for the group , see (4.14). Finally, we have
The main idea of the described decomposition process is that we are able to get a predicted decomposition of and then to extend this to so that the corresponding decomposition of the regulator of would not be changed.
Case III. Our object now is extending the decomposition (4.15) from to . Again, we start with the main decomposition
having the rigid summand .
We take the groups , and instead of , and and apply for them the above construction which preserves the decomposition of the regulator of obtained previously.
Inductively, due to the above connection restriction we get the required decomposition of each , , in finitely many steps. The proof is completed. ∎
(b) In the proof under the natural assumption we have the equalities in the condition (3) by construction, as the rank of an arbitrary homogeneous component is equal to the number of direct summands having non-trivial intersection with this component. However, the presence of rank-one summands of leads to the required inequalities in the general case.
It is routine to prove the following theorem.
Theorem 4.8 (Indecomposability Criterion)
A proper alr-group B with
is indecomposable if and only if it is rigid and its near-isomorphism invariants satisfy the connection condition: for any partition there exist and such that and are not relatively prime
The authors are thankful to the referee for her/his many helpful comments.
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