On the quasi-primary decomposition of HK-torsion theories
The paper is devoted to the study of quasi-primary decompositions of torsion theories in the rings which derivatives. It is shown that every HK-torsion theory of the differential noetherian completely bounded ring it is an intersection of finite number of quasi-primary HK-torsion theories.
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| Опубліковано в: : | Algebra and Discrete Mathematics |
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| Дата: | 2009 |
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Інститут прикладної математики і механіки НАН України
2009
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| Цитувати: | On the quasi-primary decomposition of HK-torsion theories / M. Komarnytskyi, I. Melnyk // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 2. — С. 60–69. — Бібліогр.: 17 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859777343321538560 |
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| author | Komarnytskyi, M. Melnyk, I. |
| author_facet | Komarnytskyi, M. Melnyk, I. |
| citation_txt | On the quasi-primary decomposition of HK-torsion theories / M. Komarnytskyi, I. Melnyk // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 2. — С. 60–69. — Бібліогр.: 17 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | The paper is devoted to the study of quasi-primary decompositions of torsion theories in the rings which derivatives. It is shown that every HK-torsion theory of the differential noetherian completely bounded ring it is an intersection of finite number of quasi-primary HK-torsion theories.
|
| first_indexed | 2025-12-02T09:08:48Z |
| format | Article |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 2. (2009). pp. 60 – 69
c⃝ Journal “Algebra and Discrete Mathematics”
On the quasi-primary decomposition of
HK-torsion theories
Mykola Komarnytskyi and Ivanna Melnyk
Abstract. The paper is devoted to the study of quasi-
primary decompositions of torsion theories in the rings which deriva-
tives. It is shown that every HK -torsion theory of the differential
noetherian completely bounded ring it is an intersection of finite
number of quasi-primary HK -torsion theories.
Introduction
Primary decomposition is a presentation of the ideal (or submodule) as an
intersection of primary ideals (submodules). Recently differentially prime
and primary differential ideals are investigated. In particular, Khadjiev
and Çallıalp [8] developed a theory of differentially prime ideals in asso-
ciative and non-associative differential rings by generalizing a number of
results known for associative rings without derivations.
On the other hand, starting with 1970th torsion theory intensively
develops over an ordinary rings. Different substitutes of prime ideals ap-
peared within torsion theory. The most famous is the concept, which be-
longs to Lambeck and Michler, it is constructed by critical modules. This
theory, in particular, allows to solve the problem about generalizing the
theory of primary decomposition on broader classes of noncommutative
rings. Recall a work of Storrer, in which primary decompositions of mod-
ules are obtained as a result of application of the technique of atomic and
rationally complete modules, which in a way simplifies torsion-theoretic
approach.
2000 Mathematics Subject Classification: 16S90, 13N99.
Key words and phrases: HK-torsion theory, differential kernel functor, quasi-
primary decomposition.
M. Komarnytskyi, I. Melnyk 61
In this paper a differentially prime radical is defined as the intersection
of all differentially prime differential ideals. The notion of quasi-primary
torsion theory in a category of differential modules is introduced; the re-
striction of such torsions to its full subcategory of differentially uniform
modules are investigated. The culminating point is generalization of the
theory of quasi-primary decomposition on differentially noetherian tor-
sions in a category of differentially uniform modules [10]. For technical
reasons, some properties of #-operator for differential modules are estab-
lished. Basing on this operator #-filters are studied; they prove to be
useful when investigating the differential HK -filters.
All the rings considered in this paper are assumed to be associative
with nonzero identity, and all the modules are unitary left modules, unless
otherwise specified. The word “ideal” will be used to mean a two-sided
ideal. R−Mod and R−DMod denote the categories of left R-modules and
module homomorphisms and left differential R-modules and differential
homomorphisms respectively.
Let R be a differential ring with the set of n pairwise commutative
derivations Δ = {�1, �2, . . . , �n} and let M be a left differential module
over the differential ring R. The differential structure on the module M is
defined by the set D = {d1, d2, . . . , dn} of pairwise commutative module
derivations, consistent with the corresponding ring derivations. Assume
that at least one of the derivations from the sets Δ and D is nontrivial.
If I is a left ideal of the ring R and S ⊆ R is an arbitrary subset,
then the set (I : S) = {r ∈ R∣rS ⊆ I} is a left ideal of R. In particular,
when S = {a}, where a ∈ R, (I : a) denotes the left ideal of R given by
{r ∈ R∣ra ∈ I}. If I is a differential ideal of the differential ring R, then
(I : S) and (I : a) are differential ideals.
For a ∈ R, m ∈ M we use the following notations:
a(i1,...,in) = (�i11 ∘ . . . ∘ �inn )(a), m(i1,...,in) = (di11 ∘ . . . ∘ dinn )(a),
a(∞) = {a(i1,...,in)∣i1, i2, . . . , in ∈ ℕ ∪ {0}},
m(∞) = {m(i1,...,in)∣i1, i2, . . . , in ∈ ℕ ∪ {0}}.
For any left differential ideal I and any element a ∈ R the left ideal
(I : a(∞)) is differential and the equality ((I : a(∞)) : b(∞)) = (I : (ab)(∞))
holds for any a, b ∈ R.
In the paper a standard ring-theoretic terminology will be used, fol-
lowing [2], [9].
62 On the quasi-primary decomposition
1. Operator # and its properties
Recall from [11] that a differential of the subset X of the D-module M
is a set
X# = {x ∈ M ∣x(i1,i2,...,in) ∈ X for all i1, i2, . . . , in ∈ ℕ ∪ {0}}.
The operator ( )# preserves some algebraic structures on subsets of
the D-module.
Proposition 1. Let M and N be D-modules over Δ-ring R and let
f : M → N be a differential module homomorphism. The operator
( )# on subsets of D-module has the following properties.
1. If X is a subset of the D-module M , then X# ⊆ X and (X#)# =
X#.
2. If X is a subset of the D-module M , then X# = X if and only if
the set X is differentially closed in M .
3. If X, Y are subsets of the D-module M and X ⊆ Y , then X# ⊆ Y#.
4. If {Xi}i∈I is a family of subsets of M , then
(
∩
i∈I
Xi
)
#
=
∪
i∈I
(Xi)# and
(
∪
i∈I
Xi
)
#
=
∩
i∈I
(Xi)# .
5. If X, Y are subsets of the D-module M over the �-ring R, A is a
subset of R, then
X# + Y# = (X + Y )# and (AX)# = A#X#.
6. If X is an arbitrary subset of the D-module N and f : M → N is
a differential module epimorphism, then f−1(X#) = (f−1(X))#.
7. If X is an arbitrary subset of the D-module M and f : M → N is
a differential module homomorphism, then f(X#) ⊆ (f(X))#.
If f : M → N is injective, then the equality f(X#) = (f(X))#
holds.
Proof. See [11].
Proposition 2. If X is an arbitrary subset of the D-module M over the
Δ-ring R and a ∈ R then
(
X : a(∞)
)
#
=
(
X# : a(∞)
)
.
M. Komarnytskyi, I. Melnyk 63
Proof. Without loss of generality, we may consider ordinary differential
rings and modules. Remind that
(
X : a(∞)
)
=
{
x ∈ M
∣
∣
∣
ax ∈ X, a′x ∈ X, a′′x ∈ X, ..., a(n)x ∈ X
}
.
Denote a′ = �(a), x′ = d(x).
Suppose x ∈
(
X# : a(∞)
)
. Then ax ∈ X#, a′x ∈ X#, a′′x ∈ X#,
. . . , a(n)x ∈ X# etc. It follows
(
a(i)x
)(j) ∈ X, for any i, j ∈ ℕ ∪ {0},
in particular (ax)′ = a′x + ax′ ∈ X. But a′x ∈ X, so ax′ ∈ X. From
(a′x)′ = a′′x+a′x′ ∈ X we have a′x′ ∈ X, so (ax)′′ ∈ X implies ax′′ ∈ X.
By analogy it may be established that ax(n) ∈ X for any n ∈ ℕ∪{0}. In
the same way we may prove that a′x(i) ∈ X for any i ∈ ℕ∪{0}. Applying
induction, it is easy to ascertain that a(j)x(i) ∈ X for every i, j ∈ ℕ∪{0}.
It follows that xi ∈
(
X : a(∞)
)
, and so x ∈
(
X : a(∞)
)
#
. This proves the
inclusion
(
X# : a(∞)
)
⊆
(
X : a(∞)
)
#
.
To prove the converse inclusion, we let x ∈
(
X : a(∞)
)
#
. Then
a(j)x(i) ∈ X for any i, j ∈ ℕ ∪ {0}, in particular ax′ ∈ X, a′x′ ∈ X,
a′′x′ ∈ X, . . . , a(n)x′ ∈ X, . . . , which means that x′ ∈
(
X : a(∞)
)
. In
the same way ax(i) ∈ X, a′x(i) ∈ X, a′′x(i) ∈ X . . . , a(n)x(i) ∈ X
etc., i. e. x(i) ∈
(
X : a(∞)
)
for any i ∈ ℕ ∪ {0}. Taking into account
the above reasons and the fact that
(
a(j)x(i)
)(k)
is a sum of possible
products of derivatives from the elements a ∈ R and x ∈ M , we have
that
(
a(j)x(i)
)(k) ∈ X for every i, j, k ∈ ℕ ∪ {0}. It easily follows x ∈
(
X# : a(∞)
)
. It proves the inclusion
(
X : a(∞)
)
#
⊆
(
X# : a(∞)
)
.
Note that the previous proposition naturally follows that if N is a
submodule of M , then N# is a differential submodule of M , and if N is
a differential submodule of the D-module M , then N# = N .
2. Differential kernel functors and #-filters
Remind that a functor � : R−DMod −→ R−DMod is called a differential
kernel functor in the category R−DMod [10], if the following conditions
hold:
1. � (M) is a differential submodule of M for each M ∈ R−DMod;
2. If f ∈ DHomR (M,N), then f (� (M)) ⊆ � (N);
3. � (N) = N
∩
� (M) for every differential submodule N of the dif-
ferential module M .
64 On the quasi-primary decomposition
Kernel functors were investigated in [10], [15], [16], [17].
Let � : R − Mod → R − Mod be a kernel functor. Define a functor
�# : R − DMod → R − DMod in such a way: �# (M)
df
=(� (M))# for
each M ∈ R−Mod.
Proposition 3. The functor �# : R − DMod → R − DMod is a differ-
ential kernel functor.
Proof. �# (M) = (� (M))# is obviously a differential submodule of M
for each M ∈ R−Mod.
If f : M → N is a differential homomorphism, then
f (�# (M)) = f
(
(� (M))#
)
⊆ (f (� (M)))#,
by Proposition 1, and f (� (M))# ⊆ (� (N))# = �# (N). Hence �# is a
differential preradical in R−DMod.
Let N be a differential submodule of M . Then �# (N) = (� (N))# =
(N
∩
� (M))#, and so by Proposition 1,
(
N
∩
� (M)
)
#
= N#
∩
(� (M))# = N
∩
�# (M) .
Hence �# (N) = N
∩
�# (M).
Another example of differential kernel functor provide the functor of
differential socle, i. e. the functor, which puts in correspondence to each
differential module the sum of its differentially simple submodules.
The differential kernel functor �# defines a differential torsion theory
�# =
(
T�#
,ℱ�#
)
, where T�#
= {M ∈ R−Mod ∣�# (M) = M } is a �#-
torsion class, and ℱ�#
= {M ∈ R−Mod ∣�# (M) = 0} is a �#-torsion-
free class.
To the differential kernel functor corresponds a differential preradical
filter
F�#
=
{
I— left differential ideal of R
∣
∣R/I ∈ T�#
}
.
The order relation on the class of all differential kernel functors may
be defined by the rule: � ≤ � if and only if �(M) ⊆ �(M) for all M ∈
R−DMod.
The class of all differential kernel functors forms a complete lattice.
For every pair of differential kernel functors � and � there exists its meet
� ∧ � and join � ∨ � defined by the rule:
(� ∧ �)(M) = �(M) ∩ �(M),
(� ∨ �)(M) = �(M) + �(M)
for all M ∈ R−Mod.
Then we have define #-filter as follows: (F�)#
df
=F�#
.
M. Komarnytskyi, I. Melnyk 65
Proposition 4. Let M be a left differential R-module, �, � : R−Mod −→
R −Mod be kernel functors in R −Mod. The operator ( )# for kernel
functors has the following properties:
1. �# ≤ �;
2. (�#)# = �#;
3. �# = � if and only if the set �(M) is a differential submodule of
M ;
4. If � ≤ � , then �# ≤ �#;
5. (∧i∈I�i)# = ∧i∈I (�i)#;
6. �# ∨ �# ≤ (� ∨ �)#.
Proof. 1. It is obvious that �#(M) = (�(M))# ⊆ �(M).
2. (�#)# (M) = (�#(M))# =
(
(�(M))#
)
#
, by Proposition 1, it
equals to (�(M))# = �#(M).
3. The equality �#(M) = �(M) holds if and only if (�(M))# = �(M),
but it is possible if and only if �(M) is a differential submodule of
M .
4. If � ≤ � , then �#(M) = (�(M))# ⊆ (�(M))# = �#(M).
5. We have (∧i∈I�i)# (M) =
((
∩
i∈I �i
)
(M)
)
#
= (
∩
i∈I(�i(M)))#,
and by Propositiion 1,
(
∩
i∈I(�i(M))
)
#
=
∩
i∈I (�i (M))# =
=
∩
i∈I
(
(�i)# (M)
)
= ∧i∈I (�i)# (M).
6. (�# ∨ �#) (M) = �# (M) + �# (M) = (� (M))# + (� (M))#, and
so (� (M))# +
+ (� (M))# ⊆ (� (M) + � (M))# = (� ∨ �)# (M).
Remind that a nonempty collection ℱ of left differential ideals of the
differential ring R is said to be a differential preradical HK-filter of R
(see [3]) if the following conditions hold:
HK1. If I ∈ ℱ and I ⊆ J , where J is a left differential ideal of R, then
J ∈ ℱ ;
HK2. If I ∈ ℱ and I ∈ ℱ , then I
∩
J ∈ ℱ ;
66 On the quasi-primary decomposition
HK3. If I ∈ ℱ , then (I : a(∞)) ∈ ℱ for each a ∈ R.
If a differential preradical filter ℱ satisfies an extra condition
HK4. If I ⊆ J with J ∈ ℱ and (I : a(∞)) ∈ ℱ for all a ∈ J , then I ∈ ℱ ,
then the filter ℱ is called a differential radical HK-filter.
Proposition 5. Let ℱ be a preradical filter of the left ideals of the dif-
ferential ring R and
(
I : a(∞)
)
∈ ℱ for every I ∈ ℱ and every a ∈ R.
Then ℱ# is a preradical HK-filter of the ring R. If, in addition, ℱ is a
radical filter, then ℱ# is a radical НК-filter of the noetherian ring R.
Proof. Let I ∈ ℱ# and I ⊆ J , where I, J are left differential ideals of
R. Then there exists a left ideal K ∈ ℱ such that K# = I. Consider
the left ideal K + J of the Δ-ring R. Since K ⊆ K + J and ℱ is
a preradical filter, then K + J ∈ ℱ . Since J is differential, it holds
(K + J)# = K# + J# = K# + J = I + J = J . Hence J ∈ ℱ#. Thus the
condition НК1 holds.
Let I, J ∈ ℱ#, and K, L be left ideals of ℱ , such that K# = I, L# =
J . Then K#
∩
L# = (K
∩
L)#, by Proposition 1. Since K
∩
L ∈ ℱ ,
I
∩
J ∈ ℱ#, and НК3 is proved.
For НК2, suppose I = K# for some K ∈ ℱ and let a ∈ R. Then
(
I : a(∞)
)
=
(
K# : a(∞)
)
=
(
K : a(∞)
)
#
.
Since by assumption
(
K : a(∞)
)
∈ ℱ , we see that ℱ# satisfies the condi-
tion НК2. Hence ℱ# is a preradical filter.
Assume now that ℱ is a radical filter of left ideals, which satisfies
the condition pointed in the statement. Let J ⊆ I be left differential
ideals, where I ∈ ℱ# and for every a ∈ J
(
I : a(∞)
)
∈ ℱ#. Then there
exist left ideals K,Ka ∈ ℱ , for which I = K# and
(
I : a(∞)
)
= (Ka)#.
Since the underlying ring is noetherian, then I = Rb1 + Rb2 + ... +
Rbs for some elements b1, b2, ..., bs ∈ K. Now T =
(
I : b
(∞)
1
)
∩ ⋅ ⋅ ⋅ ∩
(
I : b
(∞)
s
)
= (Kb1)# ∩ ⋅ ⋅ ⋅ ∩ (Kbs)# = (Kb1 ∩ ⋅ ⋅ ⋅ ∩Kbs)# ∈ ℱ#. But
TI ⊇ ((Kb1 ∩ ⋅ ⋅ ⋅ ∩Kbs)K)# ∈ ℱ#. Then TI ⊆ J follows J ∈ ℱ#.
3. Quasi-prime and quasi-primary torsion theories
Remind that a pretorsion theory (torsion theory) in the category of
left R-modules is called 1-pretorsion theory (1-torsion theory), if the
corresponding preradical filter (radical filter) has the basis of principal
left ideals. Every 1-pretorsion (torsion) theory defines a set Σ (F) =
M. Komarnytskyi, I. Melnyk 67
{a ∈ R ∣Ra ∈ F}, which is a left Ore set and is multiplicatively closed.
Conversely, each subset of the ring R with the properties give above de-
fines some 1-pretorsion theory.
Let � be some 1-torsion theory in the category R − Mod. Consider
the set
Ω = {� ∈ R−Mod ∣� is a HK -torsion theory, � ∧ � = {R}}
together with the partial order defined in the usual way. It is easy to
prove that this set is inductively ordered and, by Zorn’s lemma, there
exists maximal elements in Ω. The maximal of the pretorsion theories in
Ω is called a quasi-prime HK-pretorsion theory.
The existence of quasi-prime torsion theories can also be established
by using the method of transfinite induction.
Definition 1. Quasi-prime HK-torsion theory is a quasi-prime HK-pre-
torsion theory which is a torsion theory.
Example 1. Let P be a quasi-prime ideal of R. Then S = R∖P is a
dm-system. {Ra ∣a ∈ S } is a basis of the radical filter ℰP . This filter is
a 1-filter. Every maximal filter of the ones which do not meet ℰP is a
quasi-prime filter.
Definition 2. Quasi-prime radical of the HK-torsion theory � is an in-
tersection
√
� of all quasi-prime HK-torsion theories � such that � ≤ � ,
i. e. √
� =
∩
�≥�
�.
Definition 3. Quasi-primary HK-torsion theory � is a HK-torsion the-
ory such that a quasi-prime radical of which
√
� is a quasi-prime HK-
torsion theory.
Theorem 1. Every HK-torsion theory of the differential noetherian com-
pletely bounded ring has the quasi-primary decomposition, i. e. it is an
intersection of finite number of quasi-primary HK-torsion theories, which
in fact is irreducible.
Proof. The proof follows from the above definition, propositions and some
additional reasoning.
Let � be an arbitrary HK -torsion theory in the category R − DMod
and ℱ is a corresponding differential HK -filter. Then, by Generalized
Gabriel-Maranda theorem ℱ = (ℱ�̄)#. Following [1] the torsion theory �̄
over the noetherian ring is an intersection of finite number of irreducible
torsion theories �1, . . . , �n. Now use the operator # to the equality �̄ =
68 On the quasi-primary decomposition
�1 ∧ ⋅ ⋅ ⋅ ∧ �n and see what happens to the corresponding HK -filters, we
obtain
ℱ = (ℱ�1
)# ∩ ⋅ ⋅ ⋅ ∩ (ℱ�n
)# .
Thus, to prove the theorem, it is enough to show that each of the HK -
filters (ℱ�i
)# is quasi-primary. In other words, it needs to prove that the
operator # maps irreducible torsion theories into quasi-primary. It is easy
to get, considering the fact that over a completely bounded noetherian
ring every irreducible torsion theory is prime. Due to this fact, all torsion
theories �1, . . . , �n are prime, and their #-images are quasi-prime, so are
quasi-primary.
Note that the theorem shows that a completely bounded noetherian
ring is semidefinable in the sense of Golan. It may be used to get the
quasi-primary decomposition of periodical with respect to differential tor-
sion theory differential modules over noetherian differential rings.
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Contact information
M. Komarnytskyi Algebra and Logic Department, Mechanics
and Mathematics Faculty, Ivan Franko Na-
tional University of Lviv, 1 Universytetska
Str., Lviv,79000,Ukraine
E-Mail: mykola_komarnytsky@yahoo.com
I. Melnyk Algebra and Logic Department, Mechanics
and Mathematics Faculty, Ivan Franko Na-
tional University of Lviv, 1 Universytetska
Str., Lviv, 79000, Ukraine
E-Mail: ivannamelnyk@yahoo.com
Received by the editors: 11.12.2008
and in final form 11.12.2008.
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| id | nasplib_isofts_kiev_ua-123456789-154607 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-02T09:08:48Z |
| publishDate | 2009 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Komarnytskyi, M. Melnyk, I. 2019-06-15T16:49:39Z 2019-06-15T16:49:39Z 2009 On the quasi-primary decomposition of HK-torsion theories / M. Komarnytskyi, I. Melnyk // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 2. — С. 60–69. — Бібліогр.: 17 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:16S90, 13N99. https://nasplib.isofts.kiev.ua/handle/123456789/154607 The paper is devoted to the study of quasi-primary decompositions of torsion theories in the rings which derivatives. It is shown that every HK-torsion theory of the differential noetherian completely bounded ring it is an intersection of finite number of quasi-primary HK-torsion theories. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On the quasi-primary decomposition of HK-torsion theories Article published earlier |
| spellingShingle | On the quasi-primary decomposition of HK-torsion theories Komarnytskyi, M. Melnyk, I. |
| title | On the quasi-primary decomposition of HK-torsion theories |
| title_full | On the quasi-primary decomposition of HK-torsion theories |
| title_fullStr | On the quasi-primary decomposition of HK-torsion theories |
| title_full_unstemmed | On the quasi-primary decomposition of HK-torsion theories |
| title_short | On the quasi-primary decomposition of HK-torsion theories |
| title_sort | on the quasi-primary decomposition of hk-torsion theories |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/154607 |
| work_keys_str_mv | AT komarnytskyim onthequasiprimarydecompositionofhktorsiontheories AT melnyki onthequasiprimarydecompositionofhktorsiontheories |