On strongly almost m-ω₁-pʷ⁺ⁿ-projective abelian p-groups
For any non-negative integers man dn we define the class of strongly almost m-ω₁-pʷ⁺ⁿ-projective groups which properly encompasses the classes of strongly m-ω₁-pω⁺ⁿ-projective groups and strongly almost ω₁-pʷ⁺ⁿ-projective groups, defined bythe author in Demonstr. Math. (2014) and Hacettepe J. Math....
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| Опубліковано в: : | Algebra and Discrete Mathematics |
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| Дата: | 2015 |
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Інститут прикладної математики і механіки НАН України
2015
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| Цитувати: | On strongly almost m-ω₁-pʷ⁺ⁿ-projective abelian p-groups-projective abelian p-groups / P. Danchev // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 182-202. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859499701793980416 |
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| author | Danchev, P. |
| author_facet | Danchev, P. |
| citation_txt | On strongly almost m-ω₁-pʷ⁺ⁿ-projective abelian p-groups-projective abelian p-groups / P. Danchev // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 182-202. — Бібліогр.: 12 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | For any non-negative integers man dn we define the class of strongly almost m-ω₁-pʷ⁺ⁿ-projective groups which properly encompasses the classes of strongly m-ω₁-pω⁺ⁿ-projective groups and strongly almost ω₁-pʷ⁺ⁿ-projective groups, defined bythe author in Demonstr. Math. (2014) and Hacettepe J. Math. Stat.(2015), respectively. Certain results about this new group class are proved as well as it is shown that it shares many analogous basic properties as those of the aforementioned two group classes.
|
| first_indexed | 2025-11-25T01:54:20Z |
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| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 20 (2015). Number 2, pp. 182–202
© Journal “Algebra and Discrete Mathematics”
On strongly almost m-ω1-pω+n-projective
abelian p-groups
P. Danchev
Communicated by L. A. Kurdachenko
Abstract. For any non-negative integers m and n we define
the class of strongly almost m-ω1-pω+n-projective groups which
properly encompasses the classes of strongly m-ω1-pω+n-projective
groups and strongly almost ω1-pω+n-projective groups, defined by
the author in Demonstr. Math. (2014) and Hacettepe J. Math. Stat.
(2015), respectively. Certain results about this new group class are
proved as well as it is shown that it shares many analogous basic
properties as those of the aforementioned two group classes.
1. Introduction and terminology
Let all groups considered in this paper be p-primary abelian, for some
arbitrary fixed prime p. Besides, everywhere in the text, m and n are
arbitrary integers greater than or equal to {0}. Our notions and notations
are in the most part standard and follow those from the classical books
[9], [10] and [12]. The not well-known of them will be explained below in
detail.
A class of groups that plays a major role in torsion abelian group
theory is the one consisting of all almost direct sums of cyclic groups,
introduced in [11] as follows.
2010 MSC: 20K10.
Key words and phrases: almost p
ω+n-projective groups, almost ω1-pω+n-
projective groups, strongly almost ω1-pω+n-projective groups, countable subgroups,
nice subgroups, Ulm subgroups, Ulm factors.
P. Danchev 183
The separable group G is called an almost direct sum of cyclic groups
if there is a collection C consisting of nice subgroups of G, satisfying the
following three conditions:
(1) {0} ∈ C;
(2) C is closed with respect to ascending unions, i.e., if Hi ∈ C with
Hi ⊆ Hj whenever i 6 j (i, j ∈ I) then ∪i∈IHi ∈ C;
(3) If K is a countable subgroup of G, then there is L ∈ C (that is,
a nice subgroup L of G) such that K ⊆ L and L is countable.
Furthermore, an important class of p-torsion groups is the class of
all almost pω+n-projective groups, where n > 0 is an integer, defined
in [1] and [2] like this: The group G is called almost pω+n-projective if
there exists a pn-bounded subgroup P 6 G such that G/P is an almost
direct sum of cyclic groups (note that P is necessarily nice in G because
the quotient G/P is separable). It is demonstrated there that this is
tantamount to the fact that G is isomorphic to S/B, where S is an almost
direct sum of cyclic groups and B is pn-bounded.
Using the specific nature of countable subgroups, we generalized in [2]
the last concept to the following: A group G is said to be almost ω1-
pω+n-projective if there is a countable subgroup C 6 G such that G/C is
almost pω+n-projective. Notice that such a subgroup C can be chosen to
satisfy the inequalities pω+nG ⊆ C ⊆ pωG, and thus resultantly C is of
necessity nice in G.
On the other vein, we showed in [2] also that almost ω1-pω+n-projective
groups can be characterized in a different way as follows: The group G is
almost ω1-pω+n-projective if there exists a pn-bounded subgroup H 6 G
such that G/H is the sum of a countable group and an almost direct sum
of cyclic groups. As observed, such a subgroup H need not always be nice
in G, and so in [7] was given the following definition: A group G is called
strongly almost ω1-pω+n-projective if there is a pn-bounded nice subgroup
N 6 G with G/N a sum of a countable group and an almost direct sum
of cyclic groups. Note that almost pω+n-projective groups are obviously
strongly almost ω1-pω+n-projective. Some principal results concerning
certain generalizations of strongly almost ω1-pω+n-projective groups were
established in [4], [5], [6] and [8], respectively.
On the other hand, in order to extend some classical sorts of groups,
e.g. pω+n-projective groups and ω1-pω+n-projective groups, in [3] were
introduced a few classes of groups by using a single parameter m. So, the
objective of the present article is to develop that idea to some new concepts
184 On almost m-ω1-pω+n-projective abelian p-groups
which use the term “almost”, and also to find suitable relationships
between them and the mentioned above group classes.
Definition 1.1. The group G is said to be almost m-ω1-pω+n-projective if
there is a pm-bounded subgroup A of G such that G/A is strongly almost
ω1-pω+n-projective.
In particular, if A is nice in G, then G is called strongly almost m-ω1-
pω+n-projective.
If m = 0 we obtain strongly almost ω1-pω+n-projective groups, while
we obtain strongly almost ω1-pω+m-projective groups when n = 0.
Definition 1.2. The group G is said to be weakly almost m-ω1-pω+n-
projective if there is a pm-bounded nice subgroup X of G such that G/X
is almost ω1-pω+n-projective.
Substituting m = 0 we yield almost ω1-pω+n-projective groups, while
if n = 0 we yield strongly almost ω1-pω+m-projective groups. In fact,
the first fact is trivial, while for the second one we have the following
arguments: in view of Lemma 2.16 of [1] an almost ω1-pω-projective group
is actually a sum of a countable group and an almost direct sum of cyclic
groups. Hence the definition of a strongly almost ω1-pω+m-projective
group is directly applicable, and we are set.
Definition 1.3. The group G is said to be decomposably almost m-ω1-
pω+n-projective if there is a pm-bounded subgroup S of G with the property
that G/S is a sum of a countable group and an almost pω+n-projective
group.
In particular, if S is nice in G, then G is called nice decomposably
almost m-ω1-pω+n-projective. In addition, if the sum above is direct, we
shall say that G is (nice) direct decomposably almost m-ω1-pω+n-projective.
If m = 0 we identify the sums of countable groups and almost pω+n-
projective groups. If n = 0 we unify all almost ω1-pω+m-projective groups.
As for the second part, choosing m = 0 we will again obtain the sums
of countable groups and almost pω+n-projective groups, but choosing
n = 0 we will obtain strongly almost ω1-pω+m-projective groups.
Definition 1.4. The group G is called nicely almost m-pω+n-projective
if there is a pm-bounded nice subgroup Y of G such that G/Y is almost
pω+n-projective.
Putting m = 0 we get almost pω+n-projective groups, and putting
n = 0 we get almost pω+m-projective groups. Likewise, nicely almost
P. Danchev 185
m-pω+n-projective groups are both nice decomposably almost m-ω1-
pω+n-projective and almost pω+m+n-projective. Actually, almost pω+m+n-
projective groups are groups for which there is (not necessarily nice) a
pm-bounded subgroup M and, respectively, a pn-bounded subgroup N ,
such that G/M is almost pω+n-projective, respectively, G/N is almost
pω+m-projective.
Generally, the following self containments are fulfilled (this manifestly
visualizes some immediate relationships between the new group classes):
• {strongly almost ω1-pω+n-projective groups} ⊆ {decomposably al-
most n-ω1-pω+n-projective groups}.
• {strongly almost m-ω1-pω+n-projective groups} ⊆ {weakly almost
m-ω1-pω+n-projective groups}.
• {nice decomposably almost m-ω1-pω+n-projective groups}⊆{weakly
almost m-ω1-pω+n-projective groups}.
• {nicely almost m-ω1-pω+n-projective groups} ⊆ {nice decomposably
almost m-ω1-pω+n-projective groups}.
2. Some more relationships
In this section we will prove certain basic relation properties of the
groups from the above definitions. Throughout the rest of the paper, we
once again recollect that m and n are arbitrary fixed naturals or zero.
We start with the following:
Theorem 2.1. For any group G there exists a pm-bounded subgroup K
such that G/K is almost ω1-pω+n-projective if and only if G is almost
ω1-pω+m+n-projective.
Proof. We shall first show that G is as in the necessity of the theorem
⇐⇒ there exists C 6 G such that pmC is countable with pmC ⊆ pωG
and G/C is almost pω+n-projective ⇐⇒ there exists L 6 G with pm+nL
countable and G/L is an almost direct sum of cyclic groups.
Since the second equivalence follows directly by the methods used in
the proof of Theorem 2.21 from [2] or by an immediate application of
the corresponding definitions, we will be concentrated on the first one. In
fact, if pm+nL is countable, then L = R ⊕ T , where R is countable and
pm+nT = {0}. Thus
G/L = G/(R ⊕ T ) ∼= [G/(R ⊕ pnT )]/[(R ⊕ T )/(R ⊕ pnT )]
186 On almost m-ω1-pω+n-projective abelian p-groups
being an almost direct sum of cyclic groups implies that G/(R ⊕ pnT ) is
almost pω+n-projective with C = R ⊕ pnT , so pmC = pmR is countable,
as asked for.
“⇒”. Suppose by assumption that there is a pm-bounded subgroup
K 6 G such that G/K is almost ω1-pω+n-projective. Owing to Theorem
2.25 of [2], there exists a countable (nice) subgroup C/K of G/K such that
(G/K)/(C/K) ∼= G/C is almost pω+n-projective and C/K ⊆ pω(G/K) =
[∩i<ω(piG + K)]/K. Therefore, C 6 G, C = K + L for some countable
L 6 C and C ⊆ ∩i<ω(piG + K). These conditions together imply that
pmC ⊆ L is countable and pmC ⊆ ∩i<ωpi+mG = pωG, as required.
“⇐”. Write C = X ⊕ V , where X is countable and V is pm-bounded.
Hence G/C = G/(X ⊕V ) ∼= [G/V ]/(X ⊕V )/V is almost pω+n-projective,
where (X ⊕ V )/V ∼= X is countable. Thus, in accordance with [2], G/V
is almost ω1-pω+n-projective, as desired. Moreover, (X ⊕ V )/V can be
chosen so that
pm[(X ⊕ V )/V ] = (pmX ⊕ V )/V
= (pmC ⊕ V )/V ⊆ (pωG + V )/V ⊆ pω(G/V ).
This proves the preliminary claim.
Now, we have all the information necessary to prove the full assertion.
To that aim we just will show that G is almost ω1-pω+m+n-projective
⇐⇒ there is S 6 G such that pm+nS is countable and G/S is an almost
direct sum of cyclic groups, which is precisely the stated above equivalence
(compare with points (1) and (4) in Theorem 2.21 from [2]).
Necessity. Appealing to [2], G is almost ω1-pω+m+n-projective if there is a
countable subgroup K with G/K being almost pω+m+n-projective. Thus,
again in view of [2], there exists S 6 G containing K such that G/S is
an almost direct sum of cyclic groups and pm+nS ⊆ K. The last yields
that pm+nS is countable, as required.
Sufficiency. Suppose now that there exists S 6 G such that pm+nS is
countable and G/S is an almost direct sum of cyclic groups. Therefore, the
quotient G/S ∼= (G/pm+nS)/(S/pm+nS) being an almost direct sum of
cyclic groups implies with the aid of [2] that G/pm+nS is almost pω+m+n-
projective. And since pm+nS is countable, again the application of [2]
leads to G is almost ω1-pω+m+n-projective, as desired.
Remark 1. Note that the condition pmC ⊆ pωG stated in the proof of
Theorem 2.1 was at all redundant and therefore not further used. One
of the important consequences of Theorem 2.1 is that (weakly) almost
P. Danchev 187
m-ω1-pω+n-projective groups are almost ω1-pω+m+n-projective. Likewise,
the central role of Theorem 2.1 is to demonstrate unambiguously that
the concepts in Definitions 1.1 and 1.2 are nontrivial.
Imitating Theorem 2.1, it is quite natural to ask whether or not
strongly almost m-ω1-pω+n-projective groups are exactly the strongly
almost ω1-pω+m+n-projective ones. Referring to the following statement,
this seems to be true.
Proposition 2.2. If G is a strongly almost m-ω1-pω+n-projective group,
then G is strongly almost ω1-pω+m+n-projective.
Proof. Assume that there exists a pm-bounded nice subgroup T of G
such that G/T is strongly almost ω1-pω+n-projective. Thus there is
a nice subgroup A/T of G/T with the property that pnA ⊆ T and
(G/T )/(A/T ) ∼= G/A is the sum of a countable group and an almost di-
rect sum of cyclic groups. Hence pn+mA = {0} and A is nice in G (cf. [9]),
which conditions ensure that G is strongly almost ω1-pω+m+n-projective,
as claimed.
As noted above, a question of some majority is of whether or not the
converse holds, that is, whether or not every strongly almost ω1-pω+m+n-
projective group is strongly almost m-ω1-pω+n-projective.
An other question of some interest, which immediately arises, is also
whether or not almost pω+m+n-projective groups are strongly almost m-ω1-
pω+n-projective (and, in particular, weakly almost m-ω1-pω+n-projective).
This is inspired by the fact that, taking m = 0, almost pω+n-projective
groups are themselves strongly almost ω1-pω+n-projective (cf. [7]).
In this way, we have the following weaker relationship:
Proposition 2.3. If G is an almost pω+m+n-projective group, then G is
a (direct) decomposably almost m-ω1-pω+n-projective group.
Proof. Let P 6 G such that G/P is an almost direct sum of cyclic groups
and pm+nP = {0}. Since G/P ∼= [G/pnP ]/[P/pnP ], we deduce that
G/pnP is almost pω+n-projective and hence it is a sum of a countable
group and an almost pω+n-projective group. But pm(pnP ) = {0} and so
G is decomposably almost m-ω1-pω+n-projective, as promised.
Remark 2. The converse implication is, however, not true as simple
examples show. Nevertheless, decomposably almost m-ω1-pω+n-projective
groups are eventually intermediate situated between almost pω+m+n-
projective groups and almost m-ω1-pω+n-projective groups.
188 On almost m-ω1-pω+n-projective abelian p-groups
For separable groups (i.e., groups without elements of infinite height)
all of the above notions are tantamount; we do not consider here concrete
examples to show that these concepts are independent for lengths beyond
ω, but we refer the interested reader to [4], [5] or [6] for more details when
the group length is > ω.
Theorem 2.4. Suppose G is a group such that pωG = {0}. Then all of
the next points are equivalent:
(a) G is almost ω1-pω+m+n-projective;
(b) G is almost m-ω1-pω+n-projective;
(c) G is strongly almost m-ω1-pω+n-projective;
(d) G is weakly almost m-ω1-pω+n-projective;
(e) G is decomposably almost m-ω1-pω+n-projective;
(f) G is nice decomposably almost m-ω1-pω+n-projective;
(g) G is nicely almost m-pω+n-projective;
(h) G is almost pω+m+n-projective.
Proof. Apparently, all of the points (b)-(h) imply (a) and, in virtue of [2],
we obtain that point (a) holds provided (h) is fulfilled. Moreover, it is easy
to see that clause (g) implies all other ones. So, what remains to show
is the implication (h) ⇒ (g). To this purpose, [2] helps us to write that
G/Z is an almost direct sum of cyclic groups for some subgroup Z 6 G
which is bounded by pm+n. Thus (G/Z[pm])/(Z/Z[pm]) ∼= G/Z being an
almost direct sum of cyclic groups guarantees again by [2] that G/Z[pm]
is almost pω+n-projective since Z/Z[pm] ∼= pmZ is obviously bounded
by pn. But Z[pm] = Z ∩ G[pm] and both Z and G[pm] are nice in G
because G/Z is pω-bounded and G/G[pm] ∼= pmG ⊆ G is pω-bounded too.
So, resulting, Z[pm] must be nice in G (see, e.g., [9]), and since Z[pm] is
pm-bounded, we consequently get the desired fact that G is nicely almost
m-pω+n-projective.
We now proceed with two useful necessary and sufficient conditions
which are needed for applicable purposes in the next section.
Proposition 2.5. The group G is strongly almost m-ω1-pω+n-projective
if and only if there exists a pm-bounded nice subgroup T of G such that
G/(T + pω+nG) is almost pω+n-projective and pω+n(G/T ) is countable.
P. Danchev 189
Proof. It follows directly from [2] because the isomorphism
[G/T ]/pω+n(G/T ) ∼= G/(T + pω+nG)
is fulfilled.
Proposition 2.6. The group G is weakly almost m-ω1-pω+n-projective
if and only if there exists a pm-bounded nice subgroup X of G such that
G/(X+pω+nG) is almost ω1-pω+n-projective and pω+n(G/X) is countable.
Proof. It follows immediately from [2] since the isomorphism
[G/X]/pω+n(G/X) ∼= G/(X + pω+nG)
holds.
3. Ulm subgroups and Ulm factors
In [7] it was proved that if the group G is strongly almost ω1-pω+n-
projective, then so is G/pαG for any ordinal α. Here we will give a simpler
proof to the same fact devoted to almost ω1-pω+n-projective groups (see
Proposition 2.13 (b) from [2], too).
Proposition 3.1. If G is an almost ω1-pω+n-projective group, then
G/pαG is an almost ω1-pω+n-projective group for every ordinal α.
Proof. For finite ordinals α, the assertion is self-evident. So, we will assume
that α is infinite. By virtue of Theorem 2.21 (2) in [2], let G/A be the sum
of a countable group and an almost direct sum of cyclic groups for some
A 6 G with pnA = {0}. Thus, by utilizing the methods in [1] and [2], we
deduce that pα(G/A), being contained in a countable summand of G/A,
remains countable and [G/A]/pα(G/A) is again a sum of a countable
group and an almost direct sum of cyclic groups. If T ⊆ pα(G/A), the
same is still true for (G/A)/T ; we specially take T = (pαG + A)/A.
But the following isomorphisms hold:
[G/A]/(pαG + A)/A ∼= G/(pαG + A) ∼= [G/pαG]/(pαG + A)/pαG.
Observing that pn((pαG + A)/pαG) = {0}, we are finished.
Remark 4. Reciprocally, we showed in Theorem 2.16 of [2] that a group
G is almost ω1-pω+n-projective if and only if pω+nG is countable and
G/pω+nG is almost ω1-pω+n-projective.
190 On almost m-ω1-pω+n-projective abelian p-groups
Our further work in this section will be focussed on the behavior of
the new group classes about Ulm subgroups and Ulm factors. Our main
results presented below settle this matter in some aspect.
The following claim on niceness is pivotal. Its proof, although not
difficult, is rather technical, so that we leave it to the interested readers.
Lemma 3.2. Suppose N is a nice subgroup of a group A and M ⊆ pλA
for some infinite ordinal λ where pλA is bounded. Then (N + M)/M is
nice in A/M .
Lemma 3.3. Suppose that A is a group with a subgroup B such that
A/B is bounded. Thenthe following are true:
(a) If N is nice in B, then N is nice in A.
(b) If M is nice in A, then M ∩ B is nice in B.
Proof. Appealing to [9], note that a subgroup V of a group W is nice if,
for any limit ordinal δ, the equality ∩α<δ(V + pαW ) = V + pδW .
(a) Since pjA ⊆ B for some j ∈ N and hence pωA = pωB, it suffices
to check the equality only for the ordinal ω. In fact,
∩i<ω(N + piA) = ∩j6i<ω(N + piA) ⊆ ∩k<ω(N + pkB)
= N + pωB ⊆ N + pωA,
as required.
(b) We subsequently deduce that
∩α<δ(M ∩ B + pαB) ⊆ ∩α<δ(M + pαA) ∩ B = (M + pδA) ∩ B
= (M + pδB) ∩ B = M ∩ B + pδB,
as required, where the last equality follows by the modular law.
We now proceed by proving with the next crucial statement, needed
for our further application.
Proposition 3.4. Let A be a group and λ > ω an ordinal.
(i) If A is strongly almost ω1-pω+n-projective and Z ⊆ pλA, where pλA
is bounded, then A/Z is strongly almost ω1-pω+n-projective.
(ii) If X ⊆ pω+nA, pω+nA is countable and A/X is strongly almost ω1-
pω+n-projective, then A is also strongly almost ω1-pω+n-projective.
P. Danchev 191
Proof. (i) Let Q be a nice subgroup of A with pnQ = {0} and suppose
A/Q is the sum of a countable group and an almost direct sum of cyclic
groups, say A/Q = K + S. It is easily seen that Q′ = (Q + Z)/Z is
pn-bounded and in accordance with Lemma 3.2 it is nice in A′ = A/Z
as well. In addition, A′/Q′ ∼= A/(Q + Z) ∼= [A/Q]/[(Q + Z)/Q] and
(Q + Z)/Q ⊆ (Q + pλA)/Q = pλ(A/Q). Since K ∩ S ⊆ S is countable,
there exists a countable nice subgroup C of S such that K ∩ S ⊆ C.
Consequently, (A/Q)/C = [(K +C)/C]⊕[S/C]. Since S/C is pω-bounded,
we derive that
(pλ(A/Q) + C)/C ⊆ pλ((A/Q)/C) = pλ((K + C)/C)
is countable, whence so is pλ(A/Q). Furthermore, in virtue of Lemma
2.16 from [1], we observe that A/Q is actually almost ω1-pω-projective.
Since pλ(A/Q) is countable, we obtain the same for (Q + Z)/Q and thus
in accordance with Theorem 2.23 of [2], we conclude that A′/Q′ is also
ω1-pω-projective, as required.
(ii) With the aid of [7] we observe that the quotient
[A/X]/pω+n(A/X) = [A/X]/[pω+nA/X] ∼= A/pω+nA
is almost pω+n-projective. We next again employ [7] to derive that A is
strongly almost ω1-pω+n-projective, as asserted.
The next statement is pivotal.
Lemma 3.5. Suppose that A is a group with a subgroup B such that
A/B is bounded. Then
(i) A is almost pω+n-projective if and only if B is almost pω+n-projective.
(ii) A is strongly almost ω1-pω+n-projective if and only if B is strongly
almost ω1-pω+n-projective.
(iii) A is (strongly) almost m-ω1-pω+n-projective if and only if B (strongly)
almost m-ω1-pω+n-projective.
Proof. (i) It is straightforward.
(ii) Since ptA ⊆ B for some t ∈ N, we obtain that pωA = pωB
and thus pω+nA = pω+nB. Moreover, in virtue of (i), B/pω+nB =
B/pω+nA is almost pω+n-projective uniquely when A/pω+nA is almost
pω+n-projective, because the factor-group (A/pω+nA)/(B/pω+nA) ∼= A/B
remains bounded. We finally apply [7] to conclude the claim.
192 On almost m-ω1-pω+n-projective abelian p-groups
(iii) “⇒”. Let A/H be strongly almost ω1-pω+n-projective for some
H 6 A[pm] (which is nice in A). Since [A/H]/[(B + H)/H] ∼= A/(B + H)
remains bounded as an epimorphic image of A/B, we deduce with the
help of (ii) that (B + H)/H ∼= B/(B ∩ H) is strongly almost ω1-pω+n-
projective. In addition, B ∩ H 6 B[pm] (which is nice in B), and we are
finished.
“⇐”. Let B/L be strongly ω1-pω+n-projective factor-group for some
L 6 B[pm] (which is nice in B). Since [A/L]/[B/L] ∼= A/B is bounded,
point (ii) is applicable to infer that A/L is strongly almost ω1-pω+n-
projective. But L 6 A[pm] (which is nice in A), and we are done.
The niceness in both directions follows immediately from Lemma 3.3.
We have now at our disposal all the ingredients needed to prove
the following basic assertion on both Ulm subgroups and Ulm factors
pertaining to the other remaining group classes.
Proposition 3.6. If the group G is either
(a) strongly almost m-ω1-pω+n-projective or
(b) weakly almost m-ω1-pω+n-projective or
(c) nice direct decomposably almost m-ω1-pω+n-projective or
(d) nicely almost m-pω+n-projective,
then the same are both pαG and G/pαG for any ordinal α.
Proof. (a) Suppose that G/T is strongly almost ω1-pω+n-projective for
some nice pm-bounded subgroup T of G. Thus
pαG/(pαG ∩ T ) ∼= (pαG + T )/T = pα(G/T )
is also strongly almost ω1-pω+n-projective in view of [7], with pαG ∩ T
being pm-bounded and nice in pαG (cf. [9]). Hence pαG is strongly almost
m-ω1-pω+n-projective as well.
To show the second part, we consequently apply again [7] to infer that
(G/T )/pα(G/T ) = (G/T )/(pαG + T )/T ∼= G/(pαG + T )
∼= (G/pαG)/(pαG + T )/pαG
is also strongly almost ω1-pω+n-projective. Moreover, it is plainly ob-
served that (pαG + T )/pαG is bounded by pm because so is T , and that
P. Danchev 193
(pαG + T )/pαG is nice in G/pαG since it is well known that pαG + T is
nice in G - see, for example, [9].
(b) Suppose G/X is almost ω1-pω+n-projective for some nice X 6 G
with pmX = {0}. Observe that the following relations are valid:
pαG/(pαG ∩ X) ∼= (pαG + X)/X ⊆ G/X.
But a subgroup of an almost ω1-pω+n-projective group is again almost
ω1-pω+n-projective (cf. [2]). Thus pαG/(pαG ∩ X) is almost ω1-pω+n-
projective as well. Moreover, pαG ∩ X is obviously pm-bounded and
also, in accordance with [9], it is nice in pαG. So, pαG is weakly almost
m-ω1-pω+n-projective.
Furthermore,
(G/X)/pα(G/X) = (G/X)/(pαG + X)/X ∼= G/(pαG + X)
∼= (G/pαG)/(pαG + X)/pαG
is almost ω1-pω+n-projective too, owing to Proposition 3.1.
Besides, it is obviously seen that
pm((pαG + X)/pαG) = (pα+mG + pαG)/pαG = {0},
and in the case of niceness that (pαG+X)/pαG is nice in G/pαG because
it is well known that pαG + X is nice in G, see [9], for instance.
(c) Accordingly, write G/H = B ⊕ R where B is countable and R is
almost pω+n-projective for some pm-bounded nice subgroup H of G. But
pαG/(pαG ∩ H) ∼= (pαG + H)/H = pα(G/H) = pαB ⊕ pαR,
where pαB is obviously countable and pαR is by [2] almost pω+n-projective.
Since pαG ∩ H is pm-bounded and nice in pαG (see [9]), we derive that
pαG is nice direct decomposably almost m-ω1-pω+m-projective, as stated.
Concerning the other part, the direct sum
(B/pαB) ⊕ (R/pαR) ∼= [G/H]/pα(G/H)
∼= G/(pαG + H) ∼= [G/pαG]/(pαG + H)/pαG
is again a direct sum of a countable group and an almost pω+n-projective
group, because of the obvious facts that B/pαB is countable and R/pαR
is almost pω+n-projective, where the later one exploits [2]. In this vein, it
is self-evident that (pαG + H)/pαG is bounded by pm and, in conjunction
with [9], that (pαG + H)/pαG is nice in G/pαG, as required.
194 On almost m-ω1-pω+n-projective abelian p-groups
(d) Given a pm-bounded nice subgroup Y of G such that G/Y is almost
pω+n-projective. Hence, in view of [2], pαG/(pαG ∩ Y ) ∼= (pαG + Y )/Y ⊆
G/Y is almost pω+n-projective as well, with pαG ∩ Y being pm-bounded
and nice in pαG (cf. [9]).
On the other hand,
(G/pαG)/(Y + pαG)/pαG ∼= G/(Y + pαG)
∼= (G/Y )/(Y + pαG)/Y = (G/Y )/pα(G/Y )
is almost pω+n-projective by exploiting [2]. Since (Y + pαG)/pαG ∼=
Y/(Y ∩ pαG) is pm-bounded and nice in G/pαG (see [9]), the assertion
follows.
Under some extra restrictions on α, we can say even a little more:
Proposition 3.7. If G is a nice direct decomposably almost m-ω1-pω+n-
projective group, then G/pα+mG is nicely almost m-pω+n-projective for
every ordinal α 6 ω + n. In particular, G/pω+m+nG is nicely almost
m-pω+n-projective.
Proof. By Definition 1.3, we write that G/H = B⊕R where B is countable
and R is almost pω+n-projective for some pm-bounded nice subgroup H
of G. An appeal to the proof of Proposition 3.6 (c) gives that
[G/H]/pα(G/H) ∼= G/(pαG + H) ∼= [G/pα+mG]/(pαG + H)/pα+mG.
is almost pω+n-projective with
pm((pαG + S)/pα+mG) = pα+mG/pα+mG = {0},
so that the claim follows. The final part is an immediate consequence by
taking α = ω + n.
The following somewhat supplies Proposition 3.5 listed above.
Proposition 3.8. If G is a direct decomposably almost m-ω1-pω+n-
projective group, then pαG is direct decomposably almost m-ω1-pω+n-
projective for all ordinals α. In particular, if α > ω, then pαG is almost
ω1-pω+m-projective.
In addition, if G is a nice direct decomposably almost m-ω1-pω+n-
projective group and α > ω, then pαG is strongly almost ω1-pω+m-
projective.
P. Danchev 195
Proof. Using Definition 1.3, let S 6 G[pm] such that G/S = B ⊕ R where
B is countable and R is almost pω+n-projective. If α > ω, then one sees
that pαG/(pαG ∩ S) ∼= (pαG + S)/S ⊆ pα(G/S) = K ⊕ P where K is
countable and P is pn-bounded. Hence pαG/(pαG ∩ S) is also such a
direct sum of a countable group and a pn-bounded group (which itself is
a direct sum of cyclic groups) with pm-bounded intersection S ∩ pαG, so
that pαG is almost ω1-pω+m-projective.
If now α < ω is finite, then in virtue of [2] the quotient
pαG/(pαG ∩ S) ∼= (pαG + S)/S = pα(G/S) = pαB ⊕ pαR
is again a direct sum of the countable group pαB and the almost pω+n-
projective group pαR, as needed. That is why, in both cases, pαG is direct
decomposably almost m-ω1-pω+n-projective.
The final part follows easily since S being nice in G yields that S∩pαG
is nice in pαG (cf. [9]).
We now strengthen the idea in the proof of Proposition 3.1 by the
following statement; however we cannot yet establish that, for all ordinals
α, the Ulm factor G/pαG possesses the direct decomposable almost m-
ω1-pω+n-projective property provided that the same holds for G.
Proposition 3.9. If G is a direct decomposably almost m-ω1-pω+n-
projective group, then G/pαG is direct decomposably almost m-ω1-pω+n-
projective for all ordinals α > ω + n.
Proof. Utilizing Definition 1.3, write that G/S = B ⊕ R where B is
countable and R is almost pω+n-projective for some pm-bounded subgroup
S of G.
Standardly, the following isomorphisms are true:
(G/pαG)/[(S + pαG)/pαG] ∼= G/(S + pαG) ∼= (G/S)/[(S + pαG)/S].
Moreover, (S + pαG)/S ⊆ pα(G/S) = pαB. Therefore, setting T =
(S + pαG)/S, we deduce that
(G/S)/T = (B ⊕ R)/T ∼= (B/T ) ⊕ R
is again a direct sum of a countable group and an almost pω+n-projective
group. And since pm((pαG + S)/pαG) = {0}, we are finished.
196 On almost m-ω1-pω+n-projective abelian p-groups
Remark 5. When α = ω, we know by [2] or by Theorem 2.4 that G/pωG
must be almost pω+m+n-projective and thus in virtue of Proposition 2.3
it is direct decomposably almost m-ω1-pω+n-projective. However, the
unsettled situation is when ω < α < ω + n.
Now, we are ready to establish the following:
Theorem 3.10 (First Reduction Criterion). The group G is (strongly)
almost m-ω1-pω+n-projective if and only if the following two conditions
are fulfilled:
(1) pω+m+nG is countable;
(2) G/pω+m+nG is (strongly) almost m-ω1-pω+n-projective.
Proof. “⇒”. As observed before, G is almost ω1-pω+m+n-projective, so
point (1) follows automatically appealing to [2]. Concerning point (2), it
follows immediately from Proposition 3.6(a).
“⇐”. Assume now that clauses (1) and (2) are valid. For convenience
put k = m + n. By definition, let L/pω+kG 6 G/pω+kG be a pm-bounded
subgroup such that (G/pω+kG)/(L/pω+kG) ∼= G/L is strongly almost ω1-
pω+n-projective. Thus pmL ⊆ pω+kG. Since G/L is pω+k+m-bounded, we
see that pω+n(G/L) is bounded (by p2m), and applying Proposition 3.4 (i)
to G/L, we deduce that
(G/L)/(pω+nG + L)/L ∼= G/(pω+nG + L)
is strongly almost ω1-pω+n-projective, because (pω+nG+L)/L⊆pω+n(G/L).
Putting M = pω+nG + L, it is obvious that pω+nG ⊆ M and pmM =
pω+kG. That is why, G/M is strongly almost ω1-pω+n-projective with
M 6 G satisfying the above two relations.
Furthermore, supposing that Y is a maximal pm-bounded summand
of pω+nG, so there is a direct decomposition pω+nG = X ⊕ Y and, by
what we have just shown above, the inclusions X ⊆ pω+nG ⊆ M are true.
We can without loss of generality assume that X is countable because
of the following reasons: Since pω+kG = pmX is countable, it follows
that X = K ⊕ Z where K is countable and Z is pm-bounded. Therefore,
pω+nG = K ⊕ Z ⊕ Y = K ⊕ Y ′ where Y ′ = Z ⊕ Y , as needed.
We next routinely verify that X[p] = (pω+kG)[p] and thus Y ∩pω+kG =
{0}. So, suppose H is a pω+k-high subgroup of G such that H ⊇ Y .
Now, G[p] = (pω+kG)[p] ⊕ H[p] = X[p] ⊕ H[p] together with H being
pure in G (cf. [9]) readily force that G[pm] = X[pm] ⊕ H[pm] whenever
m > 1. In fact, given g ∈ G with pmg ∈ pω+kG, we write pmg = pma
P. Danchev 197
where a ∈ pω+nG = X ⊕ Y . Then pmg = pmx for some x ∈ X, whence
g ∈ x + G[pm] ⊆ X + H[pm], as required.
Besides, X ∩ H[pm] ⊆ X ∩ H = {0} and consequently (G/pω+kG)[pm]
= (X ⊕ H[pm])/pω+kG because pω+kG = pmX ⊆ X. Since M/pω+kG ⊆
(G/pω+kG)[pm], it follows that M ⊆ X ⊕ H[pm] and hence
M = (X ⊕ H[pm]) ∩ M = X + H[pm] ∩ M
by virtue of the modular law. Substituting P = H[pm] ∩ M , we derive
that pmP = {0} and that M = X + P . In addition, M = M + pω+nG =
P + pω+nG and so G/(pω+nG + P ) ∼= (G/P )/(pω+nG + P )/P is strongly
almost ω1-pω+n-projective.
We now claim that pω+n(G/P ) is countable. In fact, pω+n(G/M) is
countable because G/M is strongly almost ω1-pω+n-projective (see [7]).
But we subsequently have that
pω+n(G/M) = pn(pω(G/M)) = pn(∩i<ω(piG + M)/M)
= pn(∩i<ω(piG + P )/M) ∼= pn(∩i<ω[(piG + P )/P ]/[M/P ])
= pn(pω(G/P )/[M/P ]) = [pω+n(G/P ) + (M/P )]/[M/P ]
= pω+n(G/P )/[M/P ]
since M/P = (pω+nG + P )/P ⊆ pω+n(G/P ). Moreover,
M/P = M/(M ∩ H[pm]) ∼= (M + H[pm])/H[pm]
= (X + H[pm])/H[pm] ∼= X/(X ∩ H[pm]) ∼= X
is countable. Finally, pω+n(G/P ) is countable as well, as claimed.
Also, because (pω+nG + P )/P 6 pω+n(G/P ), Proposition 3.4 (ii)
applied to G/P shows that G/P is strongly almost ω1-pω+n-projective
with pmP = {0}, as required.
As for the “niceness” property, it can be established as Theorem 3.12
quoted below.
Now, with Proposition 2.5 at hand, we deduce the following conse-
quence.
Corollary 3.11. Suppose that pλG is countable for some ordinal λ > ω.
Then the group G is (strongly) almost m-ω1-pω+n-projective if and only
if G/pλG is.
We henceforth have all the information to prove our next basic result.
198 On almost m-ω1-pω+n-projective abelian p-groups
Theorem 3.12 (Second Reduction Criterion). The group G is weakly
almost m-ω1-pω+n-projective if and only if
(1) pω+m+nG is countable;
(2) G/pω+m+nG is weakly almost m-ω1-pω+n-projective.
Proof. “⇒”. It follows directly from [2] together with Proposition 3.6 (b).
“⇐”. For our convenience, set k = m+n. By definition, let T/pω+kG 6
G/pω+kG be a pm-bounded nice subgroup such that
(G/pω+kG)/(T/pω+kG) ∼= G/T
is almost ω1-pω+n-projective. Thus T is nice in G (see, e.g., [9]), and
pmT ⊆ pω+kG. Applying Proposition 3.1 or Proposition 2.13 (b) in [2],
G/(T + pω+nG) ∼= [G/T ]/(T + pω+nG)/T = [G/T ]/pω+n(G/T )
is also almost ω1-pω+n-projective. Putting T ′ = T + pω+nG, we see that
G/T ′ is almost ω1-pω+n-projective and that T ′ ⊇ pω+nG remains nice in
G and pmT ′ = pmT + pω+kG = pω+kG. So, replacing hereafter T ′ with
T , we may without loss of generality assume that pω+nG 6 T .
Suppose now Y is a maximal pm-bounded summand of pω+nG; so there
exists a direct decomposition pω+nG = X ⊕ Y and thus the inclusions
X ⊆ pω+nG ⊆ T hold. We may also assume with no harm of generality
that X is countable; in fact, pω+kG = pmX is countable and therefore
we can decompose X = K ⊕ Z, where K is countable and Z is pm-
bounded (whence Z is a pm-bounded summand of pω+nG and so Z ⊆ Y ).
Consequently, it is readily checked that pω+nG = K ⊕ Y with countable
summand K, as wanted.
Next, a straightforward check shows that X[p]=(pω+kG)[p]=(pmX)[p]
and thus Y ∩ pω+kG = {0} because
(Y ∩ pω+kG)[p] = Y ∩ (pω+kG)[p] = Y ∩ X[p] = {0}.
Let us now H be a pω+k-high subgroup of G containing Y (thus H is
maximal with respect to H ∩ pω+kG = {0} with H ⊇ Y ). We now assert
that
(G/pω+kG)[pm] = (X ⊕ H[pm])/pω+kG.
In fact, as noted above, X[p] = (pω+kG)[p] and thereby X ∩ H = {0}
because
(X ∩ H)[p] = X[p] ∩ H = (pω+kG)[p] ∩ H = {0}.
P. Danchev 199
Since G[p] = (pω+kG)[p] ⊕ H[p] = X[p] ⊕ H[p] and H is pure in G (see
[9]), it plainly follows that G[pm] = X[pm] ⊕ H[pm]. To prove this, given
v ∈ G with pmv ∈ pω+kG, it suffices to show that v ∈ X ⊕ H[pm]. In
fact, pmv = pmd where d ∈ pω+nG = X ⊕ Y . Then pmd = pmx for some
x ∈ X and so pmv = pmx. Therefore,
v ∈ x + G[pm] = x + X[pm] + H[pm] ⊆ X + H[pm],
as required. So, the assertion is sustained.
Furthermore, by what we have obtained above,
T/pω+kG ⊆ (G/pω+kG)[pm] = (X ⊕ H[pm])/pω+kG
implies that T ⊆ X ⊕H[pm]; note also that X ⊆ T . Put L = T ∩H[pm] ⊆
H, so that it is clear that L ∩ pω+kG = {0}. Moreover, the modular law
ensures that
T = (X ⊕ H[pm]) ∩ T = X ⊕ (T ∩ H[pm]) = X ⊕ L.
We consequently conclude that T = pω+nG + T = pω+nG + L and
G/T = G/(pω+nG + L) is almost ω1-pω+n-projective. Observe also that
L is pm-bounded, and that L is nice in G. The first fact is trivial, as for
the second one L ∩ pω+kG = {0} easily forces that L ∩ pω+nG is nice in
pω+nG and thus it is nice in G. On the other hand, as noticed above,
pω+nG + L = T is also nice in G. According to [9], these two conditions
together imply that L is nice in G, as expected.
What remains to illustrate is that pω+n(G/L) is countable. Indeed,
we have pω+n(G/L) = (pω+nG + L)/L = T/L. Also,
T/L = T/(T ∩ H[pm]) ∼= (T + H[pm])/H[pm]
= (pω+nG + H[pm])/H[pm] ∼= pω+nG/(pω+nG ∩ H[pm]).
But as obtained above, pω+nG = X⊕Y and since Y ⊆ H, we have with the
aid of the modular law that pω+nG∩H = (X ⊕Y )∩H = (X ∩H)⊕Y = Y ,
whence pω+nG ∩ H[pm] = Y [pm]. We therefore establish that
T/L ∼= (X ⊕ Y )/Y [pm] ∼= X ⊕ (Y/Y [pm]) ∼= X ⊕ pmY = X.
Since X is shown above to be countable, so does T/L = pω+n(G/L). We
finally apply Proposition 2.6 to get the desired claim.
Mimicking the method demonstrated above, with Proposition 2.6 in
hand we can state:
200 On almost m-ω1-pω+n-projective abelian p-groups
Corollary 3.13. Let λ > ω be an ordinal such that pλG is countable.
Then the group G is weakly almost m-ω1-pω+n-projective if and only if
G/pλG is.
We now ready to establish our next reduction theorem.
Theorem 3.14 (Third Reduction Criterion). The group G is nicely
almost m-pω+n-projective if and only if
(1) pω+m+nG is countable;
(2) G/pω+m+nG is nicely almost m-pω+n-projective.
Proof. “⇒”. Clause (1) follows immediately as above.
As for clause (2), it follows directly by Proposition 3.6 (d).
“⇐”. Assume that (1) and (2) are fulfilled, so that let there exist
a nice pm-bounded subgroup A/pω+m+nG of G/pω+m+nG with A 6 G
such that G/A is almost pω+n-projective. Thus, as we have seen before,
pmA ⊆ pω+kG for k = m + n, and A is nice in G. Imitating the same
technique as in Theorems 3.10 and 3.12, we can find a pm-bounded nice
subgroup N of G such that G/N is almost pω+n-projective, and so we
complete the arguments.
Same as above, we derive:
Corollary 3.15. Let λ > ω be an ordinal for which pλG is countable.
Then the group G is nicely almost m-ω1-pω+n-projective if and only if
G/pλG is.
We will be now concentrated on nice decomposably almost m-ω1-
pω+n-projective groups, which are somewhat difficult to handle. So, we
will restrict our attention on the ideal case n = 1 by showing that the
investigation of nice decomposably almost m-ω1-pω+1-projective groups
can be reduced to these of length not exceeding ω + m + 1. Specifically,
the following holds:
Theorem 3.16 (Fourth Reduction Criterion). The group G is nice direct
decomposably almost m-ω1-pω+1-projective if and only if
(1) pω+m+1G is countable;
(2) G/pω+m+1G is nice direct decomposably almost m-ω1-pω+m+1-
projective.
P. Danchev 201
Proof. The “and only if” part follows directly as above in a combination
with Proposition 3.6 (c), respectively.
Concerning the “if” part, we set for simpleness k = m + 1. Using
the corresponding definition, suppose T/pω+kG 6 G/pω+kG is a pm-
bounded nice subgroup such that [G/pω+kG]/[T/pω+kG] ∼= G/T is a
direct sum of a countable group and an almost pω+1-projective group.
Hence T is nice in G (see, e.g., [9]), and pmT ⊆ pω+kG. Also, it is
routinely checked that [G/T ]/pω+1(G/T ) ∼= G/(T + pω+1G) is almost
pω+1-projective. Henceforth, the proof goes on imitating the same scheme
of proof as that in Theorems 3.10 and 3.12 to infer the wanted statement.
Remark 6. As observed in Proposition 3.6 (c), the necessity in Theo-
rem 3.16 is valid for any natural n. However, the sufficiency probably fails
for each other n > 1.
4. Open questions
We close the work with certain challenging problems which are worth-
while for a further study.
Problem 1. Is it true that weakly almost n-ω1-pω+m-projective groups
are almost m-ω1-pω+n-projective?
Problem 2. Are (strongly) almost m-ω1-pω+n-projective groups strongly
almost ω1-pω+m+n-projective?
Problem 3. Does it follow that nice decomposably almost m-ω1-pω+n-
projective groups are strongly almost m-ω1-pω+n-projective?
Correction. In the proof of Theorem 2.23 from [2], on lines 4 and 6 the
phrase “almost pω+n-projective” should be stated as “almost ω1-pω+n-
projective”. The omission “ω1” was involuntarily.
Acknowledgment
The author owes his sincere thanks to the referee and to the handling
editor for their expert comments.
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202 On almost m-ω1-pω+n-projective abelian p-groups
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Contact information
P. Danchev Mathematical Department,
Plovdiv University “P. Hilendarski”,
Plovdiv 4000, Bulgaria
E-Mail(s): pvdanchev@yahoo.com,
peter.danchev@yahoo.com
Received by the editors: 13.09.2013
and in final form 12.10.2015.
|
| id | nasplib_isofts_kiev_ua-123456789-155166 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-11-25T01:54:20Z |
| publishDate | 2015 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Danchev, P. 2019-06-16T10:08:27Z 2019-06-16T10:08:27Z 2015 On strongly almost m-ω₁-pʷ⁺ⁿ-projective abelian p-groups-projective abelian p-groups / P. Danchev // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 182-202. — Бібліогр.: 12 назв. — англ. 1726-3255 2010 MSC:20K10. https://nasplib.isofts.kiev.ua/handle/123456789/155166 For any non-negative integers man dn we define the class of strongly almost m-ω₁-pʷ⁺ⁿ-projective groups which properly encompasses the classes of strongly m-ω₁-pω⁺ⁿ-projective groups and strongly almost ω₁-pʷ⁺ⁿ-projective groups, defined bythe author in Demonstr. Math. (2014) and Hacettepe J. Math. Stat.(2015), respectively. Certain results about this new group class are proved as well as it is shown that it shares many analogous basic properties as those of the aforementioned two group classes. The author owes his sincere thanks to the referee and to the handling editor for their expert comments. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On strongly almost m-ω₁-pʷ⁺ⁿ-projective abelian p-groups Article published earlier |
| spellingShingle | On strongly almost m-ω₁-pʷ⁺ⁿ-projective abelian p-groups Danchev, P. |
| title | On strongly almost m-ω₁-pʷ⁺ⁿ-projective abelian p-groups |
| title_full | On strongly almost m-ω₁-pʷ⁺ⁿ-projective abelian p-groups |
| title_fullStr | On strongly almost m-ω₁-pʷ⁺ⁿ-projective abelian p-groups |
| title_full_unstemmed | On strongly almost m-ω₁-pʷ⁺ⁿ-projective abelian p-groups |
| title_short | On strongly almost m-ω₁-pʷ⁺ⁿ-projective abelian p-groups |
| title_sort | on strongly almost m-ω₁-pʷ⁺ⁿ-projective abelian p-groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155166 |
| work_keys_str_mv | AT danchevp onstronglyalmostmω1pwnprojectiveabelianpgroups |