On the cotypeset of torsion-free abelian groups
In this paper the cotypeset of some torsion-free abelian groups of finite rank is studied. In particular, we determine the cotypeset of some rank two groups using the elements of their typesets.
Збережено в:
| Опубліковано в: : | Algebra and Discrete Mathematics |
|---|---|
| Дата: | 2015 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2015
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/154264 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the cotypeset of torsion-free abelian groups / F. Karimi // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 200-212. — Бібліогр.: 14 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859975474462064640 |
|---|---|
| author | Karimi, F. |
| author_facet | Karimi, F. |
| citation_txt | On the cotypeset of torsion-free abelian groups / F. Karimi // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 200-212. — Бібліогр.: 14 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | In this paper the cotypeset of some torsion-free abelian groups of
finite rank is studied. In particular, we determine the cotypeset
of some rank two groups using the elements of their typesets.
|
| first_indexed | 2025-12-07T16:23:17Z |
| format | Article |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 19 (2015). Number 2, pp. 200–212
© Journal “Algebra and Discrete Mathematics”
On the cotypeset of torsion-free abelian groups
Fatemeh Karimi
Communicated by D. Simson
Abstract. In this paper the cotypeset of some torsion-free
abelian groups of finite rank is studied. In particular, we determine
the cotypeset of some rank two groups using the elements of their
typesets.
Introduction
One of the important and known tools in the theory of torsion-free
abelian groups is type and the typeset of a group. This set which is
determined from the beginning of the the study the torsion-free groups,
has allocated many papers which are about the identifying this set for
torsion-free groups or applying it to determine the properties of these
groups and the rings over them. Problems in this area are very diverse;
for example, [3] is devoted to a determination of the representation type
of indecomposables in the categories of almost completely decomposable
groups, or in [6], the author is tried to construct indecomposable group
with an special critical typeset, and some articles as well as [4], which
are discussed about the representation of some categories of torsion-free
abelian groups, are some of the works, which are done related to type.
Moreover, [2], that provides perspectives on classification of almost com-
pletely decomposable groups and deals with the rank, regulator quotient
and near-isomorphism types, is one of the major sources in [11], which
is dealing with indecomposable (1, 2)−groups with regulator quotient of
2010 MSC: 20K15.
Key words and phrases: typeset, cotypeset, type sequence, co-rank.
F. Karimi 201
exponent 6 3 and shows that there are precisely four near-isomorphism
types of indecomposable groups. After much theorizing has been done
about the type and continued more or less to the present, another concepts
named “cotype” and “cotypeset” associated to the torsion-free groups. In
fact, The study of cotypeset of torsion-free abelian groups begins mainly
by Schultz [12]. This concept has been a focus of study between the
years 1977 to 1987, and some of the works in this area are Arnold and
Vinsonhaler [5], Metelli [9] and Mutzbauer[10]. In the past two decades
there are only a few researches about this subject, such as Lafleur [8]
in 1994. From that time better identification of cotypeset for different
groups and its relation with type is considered. Moreover, always such a
question is raised that: could we have some results for cotype similar to
ones about the type? For example, similar results of [3], [4], [11] or [2]
could be stated for cotype instead of types?
In this paper, we deal with the cotypeset of some torsion-free abelian
groups of finite rank and show that the cotypeset of any completely
decomposable group is closed under mutually union of its rank one direct
summand’s types. Moreover, we have some results about the relation
between the elements of cotypeset and typeset and determine the cotypeset
of some rank two groups using their typesets.
Finally, some of the other unsolved problems in this area are as follows:
(1) Identifying the cotypeset for a completely decomposable group of
rank greater than 2.
(2) If A = A1 ⊕ A2 is a group of rank three, with r(A2) = 2, then can
we obtain CT(A), (the cotypeset of A) using the cotypesets of A1
and A2?
(3) Is there any relation between the cardinality of the cotypeset of a
torsion-free abelian group and the existance of a non-zero ring on a
group?
1. Notation and Preliminaries
All groups considered in this paper are torsion-free and abelian, with
addition as the group operation. Terminology and notation will mostly
follow from [7]. By the typeset of a torsion-free group A we mean the
partially ordered set of types, i.e.,
T(A) = {t(x) | 0 6= x ∈ A},
202 Cotypeset of torsion-free groups
and for two types t1 = [(mi)i∈N] and t2 = [(ki)i∈N] we define:
inf{t1, t2} = [(min{mi, ki})i∈N], sup{t1, t2} = [(max{mi, ki})i∈N].
Moreover, if t2 6 t1 then we set
t1 − t2 = [(mi − ki)i∈N].
We also may use the notations t1 ∩ t2 and t1 ∪ t2 instead of inf{t1, t2} and
sup{t1, t2} respectively, for more convenience. A pure subgroup B of A is
said to be of co-rank one if rank(A/B) = 1. The cotypeset of A, denoted
by CT(A), is defined as
CT(A) = {t(A/B) | B is a pure co-rank one subgroup of A}.
A torsion-free group A is called cohomogeneous if CT(A) has cardinality
equal to one.
Let A is a torsion-free group of rank n and S = {x1, x2, . . . , xn} a
maximal independent set of A. For Xi = 〈xi〉∗ and
Yi = 〈x1, x2, . . . , xi−1, xi+1, . . . , xn〉∗,
define the inner type of A to be
IT(A) = inf{t(X1), . . . , t(Xn)}.
Moreover, the outer type of A is as follows
OT(A) = sup{t(A/Y1), . . . , t(A/Yn)}.
2. Cotypeset of rank two groups
As in [5], let A is a rank two group and A1, A2, · · · be an indexing
of the pure rank one subgroups of A with ti = t(Ai), σi = t(A/Ai) for
each i. Define TA = (t1, t2, · · · ), CTA = (σ1, σ2, · · · ) are two countable
infinite sequences of types (repetition of types is allowed). We say two
type sequences T and T ′ are equivalent, T ≈ T ′, if one is a permutation
of the other, and by this, TA and CTA are unique up to equivalence.
F. Karimi 203
Proposition 1. Let A be a rank 2 group with TA = (t1, t2, · · · ) and
CTA = (σ1, σ2, · · · ).
(1) There is a type t0 such that t0 = inf{ti, tj} for each i 6= j and if
T (A) is finite then t0 = ti for some i > 1.
(2) There is a type σ0 such that σ0 = sup{σi, σj} for each i 6= j and if
CT (A) is finite then σ0 = σi for some i > 1.
(3) ti 6 σj for each i 6= j and t0 6 σ0.
(4) σi − tj = σj − ti for each i 6= j with i > 0 and j > 0.
(5) If t0 = t(Z) then σi = σ0 − ti for each i.
Proof. See ([5], Proposition 1.1).
Using above Proposition and nothing the known fact from [13], which
the typeset of any non-nil rank two torsion-free group has the cardinality
at most three, it would be straight forward too check:
Proposition 2. The cotypeset of a non-nil rank two torsion-free group
A has one of this forms:
(1) If T(A) = {t} and B is a pure subgroup of A with t(B) = t, then
CT(A) = {t(A/B)}.
(2) If T(A) = {t1, t2} with t1 < t2 and A1 is a pure subgroup of
A such that t(A1) = t1, then CT(A) = {σ1, σ2} such that σ1 =
t(A/A1), σ2 = σ1 − t2 + t1.
(3) If T(A) = {t0, t1, t2} with t0 < t1, t2 and A1, A2, A3 are rank one
pure subgroups of A in which t(A1) = t1, t(A2) = t2, t(A3) = t3, then
CT(A) = {σ1, σ2, σ3} such that σ3 = t(A/A3), σ1 = σ3+t0−t1, σ2 =
σ3 + t0 − t2.
Moreover, we could easily show that:
Corollary 1. If A is a non-nil rank two group which is completely de-
composable, then we have:
(1) If |T(A)| = 1 or 2, then T(A) = CT(A).
(2) If T(A) = {t1, t2, t1 ∩ t2}, then CT(A) = {t1, t2, t1 ∪ t2}.
Lemma 1. Let T = (t1, t2, · · · ) and C = (σ1, σ2, · · · ) be type sequences
with t0 = inf{ti, tj} and σ0 = sup{σi, σj} whenever i 6= j. There is a
rank two group A with TA = T and CTA = C if and only if there is a
rank two group B with TB = (t1 − t0, t2 − t0, · · · ), CTB = (σ1 − t0, σ2 −
t0, · · · ), IT(B) = t(Z), OT(B) = σ0 − t0.
204 Cotypeset of torsion-free groups
Proof. See ([5], Lemma 1.3).
Proposition 3. Let S = {ti | i > 1} be a set of types with t0 = inf{ti, tj}
whenever i 6= j. If there exists characteristic hi ∈ ti for i > 0 with
h0 = inf{hi, hj} for each i 6= j then there exists a rank two group A with
T (A) = S and OT(A) = [sup{hi | i > 1}].
Proof. See ([5], Corollary 2.14).
Theorem 1. Let S = {t1, t2, · · · } be a set of types with t0 = inf{ti, tj}
for each i 6= j and t0 ∈ S if S is finite. Then
(1) There exists si ∈ ti for i > 0 such that s0 = min{si, sj} for i 6= j.
(2) There exists a rank two group A with T (A) = S, IT(A) = t0,
OT(A) = [sup{si | i > 1}] and CT (A) = {OT(A)−(ti −t0) | i > 1}.
Proof. (1) Let n > 3 be an arbitrary integer and let si ∈ ti for 0 6 i 6 n−1
with s0 = min{si, sj} for 1 6 i 6= j 6 n − 1. Now choose sn ∈ tn such
that s0 = min{si, sn} for 1 6 i 6 n − 1.
(2) By (1) and Proposition 3, let χ′
0 = sup{si | i > 1}, σ′
0 = [χ′
0]
and γi = σ′
0 − ti for i > 0. Note that γi = [χ′
0 − si] for each i > 0. Now
Γ = {γ1, γ2, · · · } with γ0 = sup{γi, γj} if i 6= j, because t0 = inf{ti, tj}
hence σ′
0 −t0 = sup{σ′
0 −ti, σ′
0 −tj}. Moreover, γ0 ∈ Γ if Γ is finite. In fact
if Γ is finite then S must be finite. This means t0 ∈ S which yields t0 = tj
for some tj ∈ S. Now we have γ0 = σ′
0 −t0 = σ′
0 −tj = γj , for some γj ∈ Γ.
Define σi = γ0 − γi for i > 0. The next step is to show that there exists a
rank two group B with T (B) = {σi | i > 1} and CT (B) = {γi | i > 1}.
For each i > 1, let χi = (χ′
0 − s0) − (χ′
0 − si) ∈ σi = γ0 − γi. Note that
1) If χi(p), the p−component of χi, is equal to ∞, for some i > 1, then
si(p) = ∞, s0(p) < ∞ and χ′
0(p) = ∞.
2) χi(p) = si(p) − s0(p).
3) min{χi, χj} = (0, 0, · · · ) whenever i 6= j. This is a consequence of 2)
and the fact that s0 = min{si, sj}.
By 3) and Proposition 3, there exists a rank two group B with T (B) =
{σi | i > 1}, OT(B) = [sup{χi | i > 1}]. Moreover, from 3) we deduce
that IT(B) = t(Z). Now 2) implies
sup{χi | i > 1} = sup{si − s0 | i > 1}
= sup{si | i > 1} − s0
= χ′
0 − s0,
F. Karimi 205
therefore OT(B) = γ0. Now by Proposition 1 (5), we deduce
CT (B) = {γ0 − σi | i > 1} = {γi | i > 1}.
The last equality holds because of 3). In fact:
γ0 − σi = [(χ′
0 − s0) − (si − s0)] = [χ′
0 − si] = γi.
Consequently, in view of Lemma 1, there exists a rank two group A with
T (A) = {σi + t0 | i > 1} = {ti | i > 1}, IT(A) = t0,
CT (A) = {γi + t0 | i > 1} = {σ′
0 − ti + t0 | i > 1},
OT(A) = OT(B) + t0 = γ0 + t0 = σ′
0.
3. Cotypeset of finite rank groups
We begin this section with an example of a cohomogeneous group
of any arbitrary finite rank that is homogeneous too. First we need the
following definition and two propositions:
Definition 1. A torsion-free group A is called coseparable if, given any
pure subgroup B of A such that A/B reduced of finite rank, B contains
a summand C of A which has a completely decomposable finite rank
complement. Moreover, a torsion-free group A is finitely cohesive exactly
if for every pure finite corank subgroup B of A, A/B is divisible.
Proposition 4. A finite rank group is coseparable exactly if it is com-
pletely decomposable.
Proof. See ([9], Proposition 1.2).
Remark 1. By above definition, a finitely cohesive group A is cohomo-
geneous with
CT(A) = {(∞, ∞, · · · )}.
Proposition 5. Finitely cohesive groups are coseparable.
Proof. See ([9], Proposition 1.5).
Example 1. Let A be a finitely cohesive group of finite rank. Then by
Proposition 5, A is coseparable and so completely decomposable group
by Proposition 4. This yields T(A) = {(∞, ∞, · · · )} and so A is a homo-
geneous group.
206 Cotypeset of torsion-free groups
Now we present the main results of this section.
Theorem 2. Let A is a torsion-free group of finite rank n, A set
{x1, x2, · · · , xn} a maximal independent set of A and A1, A2, · · · is an
indexing of the rank one pure subgroups of A. Define
UA = {m1x1 + · · · + mnxn | m1, m2, · · · , mn ∈ Z, (m1, m2, · · · , mn) = 1}
which is a subset of
⊕n
i=1 Zxi ⊆ A. Then
(1) For each i > 1 there exists a unique ai ∈ UA
⋂
Ai. Moreover,
Ai
⋂
(
n
⊕
i=1
Zxi) = Z(ai), t(ai) = t(Ai).
(2) OT(A) = [sup{χA(a) | a ∈ UA}].
Proof. (1) Let a′
i be a non-zero element of Ai with t(a′
i) = ti = t(Ai).
Then Ai = 〈a′
i〉∗ and Ai
⋂
UA 6= 0. Now for all i > n, let k, ki1, ki2, · · · , kin
be some integers such that
0 6= ka′
i =
n
∑
j=1
kijxj .
Suppose (ki1, ki2, · · · , kin) = l; if l = 1 then ka′
i has the stated properties
in (1). If l 6= 1 ∈ Z then we could write kij = lk′
ij , (j = 1, 2, · · · , n)
and ka′
i = l(
∑n
j=1 k′
ijxj). Now by letting ai =
∑n
j=1 k′
ijxj we obtain
ai ∈ UA
⋂
Ai. To show that ai is unique, let a′′
i ∈ UA
⋂
Ai, then from
ai, a′′
i ∈ Ai, there exist some integers m, n such that (n, m) = 1 and
ma′′
i = nai. Now using the fact that a′′
i , ai ∈ UA we conclude the result
and the other parts of (1) are easy to proof.
(2) We write
(A/
n
⊕
i=1
Zxi) =
⊕
p
[Z(pi1p) ⊕ · · · ⊕ Z(pinp)]
such that 0 6 i1p 6 · · · 6 inp 6 ∞ for each p. Then IT(A) = [(i1p)]
and OT(A) = [(inp)], (See [14]). If a + (
⊕n
i=1 Zxi) is an element of the
p−component of A/
⊕n
i=1 Zxi, then the order of a + (
⊕n
i=1 Zxi) is the
least j such that pja = mu for some u ∈ UA and m ∈ Z with (m, p) = 1.
Since inp is the maximum of such j, in view of j 6 hA
p (u), we have
inp 6 sup{hA
p (a) | a ∈ UA}.
F. Karimi 207
But
A
⊕n
i=1 Zxi
⊇
Ai + (
⊕n
i=1 Zxi)
⊕n
i=1 Zxi
∼=
Ai
Zai
=
⊕
p
Z(plp),
such that inp > lp = hA
p (ai). This means sup{hA
p (a) | a ∈ UA} 6 inp and
therefore
OT(A) = [(inp)] = [sup{χA(a) | a ∈ UA}].
Theorem 3. Let A is a torsion-free group of finite rank n and A1,
A2, · · · , An are rank one subgroups such that {xi|xi ∈ Ai}
n
i=1 is an
independent set of A. If σi = t
(
A
〈
⊕n
i6=j=1
Aj〉∗
)
and t(Ai) = ti, then
σi − ti = σj − tj for all i 6= j ∈ {1, 2, · · · , n}.
Proof. There is an exact sequence
0 −→ Ai −→
A
〈
⊕n
i6=j=1 Aj〉∗
−→
A
⊕n
j=1 Aj
−→ 0
for all i = 1, 2, · · · , n. Choose ai ∈ Ai and yi ∈ A
〈
⊕n
i6=j=1
Aj〉∗
with ai 7−→ yi.
Then
0 −→
Ai
Zai
−→
A/〈
⊕n
i6=j=1 Aj〉∗
Zyi
−→
A
⊕n
j=1 Aj
−→ 0 (∗)
is exact. Now since A/〈
⊕n
i6=j=1 Aj〉∗ is a rank one torsion-free group,
A/〈
⊕n
i6=j=1
Aj〉∗
Zyi
is torsion, so we have
A
⊕n
j=1 Aj
∼=
⊕
p
Z(pkp),
Ai
Zai
∼=
⊕
p
Z(plp)
and
A/〈
⊕n
i6=j=1 Aj〉∗
Zyi
∼=
⊕
p
Z(pnp).
On the other hand the exactness of (∗) implies that
⊕
p
Z(pkp) ∼=
⊕
p Z(pnp)
⊕
p Z(plp)
,
hence kp = np − lp. Moreover,
np = h
A/〈
⊕n
i6=j=1
Aj〉∗
p (yi) − hZyi
p (yi), lp = hAi
p (ai) − hZai
p (ai)
208 Cotypeset of torsion-free groups
and hZyi
p (yi) = 0 = hZai
p (ai). Therefore
kp = h
A/〈
⊕n
i6=j=1
Aj〉∗
p (yi) − hAi
p (ai),
which means [(kp)] = σi − ti. Similarly [(kp)] = σj − tj and this completes
the proof.
Proposition 6. In any torsion-free abelian group of finite rank A with
finite typeset, the intersection type and inner type coincide and this type
is realized. This means there exists a rank one subgroup B of A such that
IT(A) = t(B).
Proof. See ([10], Corollary 1.3).
Proposition 7. Let A is a group of rank two and X, Y be different pure
rational subgroups of A. Then
t(A/X) − t(Y ) = t(A/Y ) − t(X).
Moreover, the outer type is realized if the inner type is realized; more pre-
cisely if t(B) = IT(A) for some subgroup B of A then t(A/B) = OT(A).
Proof. See ([10], Lemma 2.4).
Theorem 4. Let A is a group such that any rank two torsion-free quotient
of A is non-nil. Then CT(A) is closed under the union of its elements.
Proof. Let s, t ∈ CT(A) be two arbitrary elements. Then there exist
pure subgroups B, C of A such that A/B and A/C are of rank one and
t(A/B) = s, t(A/C) = t. Now D = A
B∩C is a torsion-free group of rank
two and B
B∩C , C
B∩C are two co-rank one pure subgroups of D such that
t
(
D
B/B ∩ C
)
= s, t
(
D
C/B ∩ C
)
= t.
Hence s, t ∈ CT(D). Moreover, our assumption implies that CT(D) ⊆
CT(A). Now it is sufficient to prove s ∪ t ∈ CT(D). By Proposition 6 the
inner type of D is realized in T(D), because T(D) is finite, so there is a pure
subgroup Y of D with r(Y ) = 1, IT(D) = t(Y). Now by Proposition 7 we
have t(D/Y ) = OT(D) = s∪ t ∈ CT(D) and this completes the proof.
F. Karimi 209
Theorem 5. Let A = A1 ⊕ A2 be a group of rank three with A2 a non-nil
group of rank two. Then
CT(A) ⊇ {t(A1)} ∪ CT(A2)
and T(A) contains at most three maximal elements.
Proof. The first part is obtained from the fact that for any pure co-rank
one subgroup B of A2, A1 ⊕ B is a pure co-rank one subgroup of A such
that
A
A1 ⊕ B
∼=
A2
B
.
Moreover,
T(A) = {t(A1)} ∪ T(A2) ∪ {t(A1) ∩ t | t ∈ T(A2)}.
But T(A2) has at most two maximal elements since A2 is a non-nil rank
two group.
Remark 2. At the proof of above theorem, if Y is any pure fully invariant
subgroup of A with r(A/Y ) = 1 and Y 6= A2, then Y ∩ A2 6= 0 and
Y ∩ A1 = A1. In fact if Y ∩ A1 6= A1,
A
Y
=
A1 ⊕ A2
(Y ∩ A1) ⊕ (Y ∩ A2)
is not a torsion-free group, (because A1/(Y ∩ A1) is torsion) which yields
a contradiction.
Now 0 6= Y ∩ A2 is a pure subgroup of rank one of A2. Let pa2 = y
for some a2 ∈ A2, y ∈ Y ∩ A2 and a prime number p, then there exist an
element a ∈ Y such that a = a1 + a′
2 for some a1 ∈ A1 and a′
2 ∈ A2 in
which pa2 = y = pa = pa1 + pa′
2, because Y is a pure subgroup of A, but
this yields a2 = a′
2 and a1 = 0. Therefore a′
2 ∈ Y ∩ A2 and this completes
this part of proof. So we have Y = A1 ⊕ (Y ∩ A2) and Y ∩ A2 is a co-rank
one pure subgroup of A2.
But if Y is not a fully invariant subgroup, similar result couldn’t be
true.
210 Cotypeset of torsion-free groups
Theorem 6. Let X =
⊕n
i=1 Xi is a completely decomposable group of
rank n and B a torsion-free group with finite rank greater than one. If
A = X ⊗ B and Bi = Xi ⊗ B, then
CT(A) ⊇
n
⋃
i=1
CT(Bi)
and T(A) is equal to
n
⋃
i=1
T(Bi)
⋃
{∩i∈Iti| I is a finite subset of{1, 2, · · · , n}, ti ∈ T(Bi)}.
Proof. Let Ci is a pure rank (co-rank) one subgroup of Bi, then
Ci(
n
⊕
i6=j=1
Bj
⊕
Ci)
is a pure rank (co-rank) one subgroup of A. Moreover,
A
(
⊕n
i6=j=1 Bj)
⊕
Ci
∼=
Bi
Ci
which yields the result.
Theorem 7. If A =
⊕n
i=1 Bi with r(Bi) > 2, then the typeset of A is
equal to
n
⋃
i=1
T(Bi)
⋃
{∩i∈Iti | I is a finite subset of {1, 2, · · · , n}, ti ∈ T(Bi)}
and
CT(A) ⊇
n
⋃
i=1
CT(Bi).
Proof. Obvious.
In this part we have some results about the cotypeset of completely
decomposable groups.
Lemma 2. Let A be a torsion-free group and H 6 A then
CT (A/H) ⊆ CT (A).
Proof. Obvious.
F. Karimi 211
Theorem 8. Let A =
⊕
i∈I Ai is a torsion-free group with r(Ai) = 1 and
t(Ai) = ti. Then the cotypeset of A is closed under mutually union of
t(Ai)s. Moreover, if ti = t(Ai) and tj = t(Aj) are two incomparable types,
then (ti ∪ tj) − tk = tk − (ti ∩ tj) for k = i, j.
Proof. We know A/(⊕(j 6=)i∈IAi) ∼= Aj , hence tj ∈ CT (A) for all j ∈ I.
Now let ti, tj be two incomparable types and let A′ = Ai ⊕ Aj . Then A′
is a pure subgroup of A and from T (A′) = {ti, tj , ti ∩ tj} ⊆ T (A) and
Proposition 1, we deduce that CT (A′) = {ti, tj , ti ∪ tj}. Let ti ∪ tj =
t(A′/H) for some co-rank one subgroup H of A′. Now
A′ ⊕ (
⊕
(i,j 6=)k∈I Ak)
H ⊕ (
⊕
(i,j 6=)k∈I Ak)
is a rank one torsion-free quotient of A. We let G = H ⊕ (
⊕
(i,j 6=)k∈I Ak),
hence A/G ∼= A′/H which yields t(A/G) = t(A′/H) = ti ∪ tj ∈ CT (A).
Moreover, by assuming A′ = Ai ⊕ Aj we have OT(A′) = ti ∪ tj and
IT(A′) = ti ∩ tj . Now the result follows from Prposition 1(4) and the fact
that OT(A′) ∈ CT (A) and IT(A′) ∈ T (A).
References
[1] A.M. Aghdam and A. Najafizadeh,On torsion free rings with indecomposable
additive group of rank two, Southeast Asian Bull. Math., 32, No. 2, 2008, 199-208.
[2] D.M. Arnold, Abelian Groups and Representations of Partially Ordered Sets, CMS
Adv. Books Math., Springer-Verlag, New York, 2000.
[3] D. M. Arnold and M. Dugas, Representation type of finite rank almost completely
decomposable groups, Forum Math., 10, 1998, 729-749.
[4] D. M. Arnold and D. Simson, Representations of Finite Posets Over Discrete
Valuation Rings, Comm. Algebra, 35, Issue 10, 2007, 3128-3144.
[5] D.M. Arnold, C. Vinsonhaler,The typesets and cotypesets rank two torsion-free
abelian groups, Pacific J. Math., 114, No. 1, 1984, 1-21.
[6] M. Dugas, BCD-groups with type set (1,2), Forum Mathematicum., 13, 2000,
143-148.
[7] L. Fuchs, Infinite abelian groups, Vol. 2, Academic Press, New York - London,
1973.
[8] R. S. Lafleur, Typesets and cotypesets of finite-rank torsion-free abelian groups,
Contemp. Math., 171, Amer. Math. Soc., Providence, RI, 1994, 243-256.
[9] C. Metelli, Coseparable torsion-free groups, Arch. Math., 45, 1985, 116-124.
[10] O. Mutzbauer, Type invariants of torsion-free abelian groups, Abelian Group
Theory Proc. Perth, 1987, 133-153.
[11] O. Mutzbauer and E. Solak, (1, 2)−Groups with p
3-regulator quotient, J. Algebra,
320, 2008, 3821-3831.
212 Cotypeset of torsion-free groups
[12] P. Schultz, The type-set and cotype-set of a rank two abelian group, Pacific J.
Math., 78, 1978, 503-517.
[13] A.E. Stratton, The type-set of torsion-free rings of finite rank, Comment. Math.
Univ. Sancti Pauli, 27, 1978, 199-211.
[14] R. Warfield, Jr., Homomorphisms and duality of torsion-free groups, J. Algebra,
83 No. 2, 1983, 380-386.
Contact information
Fatemeh
Karimi
Department of Mathematics,
Payame Noor University, PO BOX 19395-3697,
Tehran, I.R. of IRAN
E-Mail(s): karimi@pnu.ac.ir
Received by the editors: 08.07.2012
and in final form 09.06.2014.
|
| id | nasplib_isofts_kiev_ua-123456789-154264 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T16:23:17Z |
| publishDate | 2015 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Karimi, F. 2019-06-15T12:04:00Z 2019-06-15T12:04:00Z 2015 On the cotypeset of torsion-free abelian groups / F. Karimi // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 200-212. — Бібліогр.: 14 назв. — англ. 1726-3255 2010 MSC:20K15 https://nasplib.isofts.kiev.ua/handle/123456789/154264 In this paper the cotypeset of some torsion-free abelian groups of finite rank is studied. In particular, we determine the cotypeset of some rank two groups using the elements of their typesets. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On the cotypeset of torsion-free abelian groups Article published earlier |
| spellingShingle | On the cotypeset of torsion-free abelian groups Karimi, F. |
| title | On the cotypeset of torsion-free abelian groups |
| title_full | On the cotypeset of torsion-free abelian groups |
| title_fullStr | On the cotypeset of torsion-free abelian groups |
| title_full_unstemmed | On the cotypeset of torsion-free abelian groups |
| title_short | On the cotypeset of torsion-free abelian groups |
| title_sort | on the cotypeset of torsion-free abelian groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/154264 |
| work_keys_str_mv | AT karimif onthecotypesetoftorsionfreeabeliangroups |