Closure operators in modules and adjoint functors, I
In the present work the relations between the closure operators of two module categories are investigated in the case when the given categories are connected by two covariant adjoint functors H: R-Mod → S-Mod and T : S-Mod → R-Mod. Two mappings are defined which ensure the transition between the clo...
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irk-123456789-1883502023-02-24T01:27:31Z Closure operators in modules and adjoint functors, I Kashu, A.I. In the present work the relations between the closure operators of two module categories are investigated in the case when the given categories are connected by two covariant adjoint functors H: R-Mod → S-Mod and T : S-Mod → R-Mod. Two mappings are defined which ensure the transition between the closure operators of categories R-Mod and S-Mod. Some important properties of these mappings are proved. It is shown that the studied mappings are compatible with the order relations and with the main operations. 2018 Article Closure operators in modules and adjoint functors, I / A.I. Kashu // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 1. — С. 98-117. — Бібліогр.: 13 назв. — англ. 1726-3255 2010 MSC: 16D90, 16S90. http://dspace.nbuv.gov.ua/handle/123456789/188350 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In the present work the relations between the closure operators of two module categories are investigated in the case when the given categories are connected by two covariant adjoint functors H: R-Mod → S-Mod and T : S-Mod → R-Mod. Two mappings are defined which ensure the transition between the closure operators of categories R-Mod and S-Mod. Some important properties of these mappings are proved. It is shown that the studied mappings are compatible with the order relations and with the main operations. |
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Kashu, A.I. |
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Kashu, A.I. Closure operators in modules and adjoint functors, I Algebra and Discrete Mathematics |
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Kashu, A.I. |
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Kashu, A.I. |
| title |
Closure operators in modules and adjoint functors, I |
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Closure operators in modules and adjoint functors, I |
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Closure operators in modules and adjoint functors, I |
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Closure operators in modules and adjoint functors, I |
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Closure operators in modules and adjoint functors, I |
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closure operators in modules and adjoint functors, i |
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Інститут прикладної математики і механіки НАН України |
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Closure operators in modules and adjoint functors, I / A.I. Kashu // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 1. — С. 98-117. — Бібліогр.: 13 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT kashuai closureoperatorsinmodulesandadjointfunctorsi |
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2025-07-16T10:22:22Z |
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2025-07-16T10:22:22Z |
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1837798621232234496 |
| fulltext |
“adm-n1” — 2018/4/2 — 12:46 — page 98 — #100
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 25 (2018). Number 1, pp. 98–117
c© Journal “Algebra and Discrete Mathematics”
Closure operators in modules
and adjoint functors, I
A. I. Kashu
Abstract. In the present work the relations between the
closure operators of two module categories are investigated in the
case when the given categories are connected by two covariant
adjoint functors H : R-Mod −→ S-Mod and T : S-Mod −→ R-Mod.
Two mappings are defined which ensure the transition between the
closure operators of categories R-Mod and S-Mod. Some important
properties of these mappings are proved. It is shown that the studied
mappings are compatible with the order relations and with the main
operations.
1. Introduction. Preliminary notions and facts
The aim of this paper is to clarify connections between the closure
operators of two module categories in the adjoint situation. For that we fix
an arbitrary (R,S)-bimodule RUS and consider the following two covariant
functors:
R-Mod
H=HomR(U,-)// S-Mod,
T=U⊗
S
-
oo
where T is left adjoint to H . We remark that any pair of covariant adjoint
functors between two module categories has such a form (up to a functorial
isomorphism). This adjoint situation is characterized by two natural
transformations (functorial morphisms):
Φ: TH → R−Mod, Ψ: S−Mod → HT,
2010 MSC: 16D90, 16S90.
Key words and phrases: category of modules, closure operator, adjoint functors,
lattice operations.
“adm-n1” — 2018/4/2 — 12:46 — page 99 — #101
A. I. Kashu 99
which satisfy the conditions:
H(ΦX) ·ΨH(X) = H(X), (1.1)
ΦT (Y ) · T (ΨY ) = T (Y ), (1.2)
for every modules X ∈ R-Mod and Y ∈ S-Mod.
This situation was studied in a series of works [1–6], where the relations
between preradicals of categories R-Mod and S-Mod are shown. The
ideas and methods used in these works can partially be adopted for
the investigation of connections between closure operators of the given
categories. This question is studied by other methods in the book [9]
(§ 5.13).
Now we recall some notions and facts which are necessary for the
following account. A closure operator of R-Mod is a mapping C which
associates to every pair N ⊆ M , where N ∈ L(M), a submodule of M
denoted by CM(N) which satisfies the conditions:
(c1) N ⊆ CM(N) (extension);
(c2) If N1, N2 ∈ L(M) and N1⊆N2, then CM(N1)⊆CM(N2) (monotony);
(c3) For every R-morphism f : M → M ′ and N ∈ L(M) we have
f
(
CM(N)
)
⊆ CM′
(
f(N)
)
(continuity),
where M ∈ R-Mod and L(M) is the lattice of submodules of M ([7–13]).
We denote by CO(R) the class of all closure operators of R-Mod. In
the class CO(R) the relation of partial order is defined as follows:
C 6 D ⇔ CM(N) ⊆ DM(N) for every N ⊆ M.
Moreover, in CO(R) the operations “∨” (join) and “∧” (meet) are defined
by the following rules:
(
∨
α∈A
Cα
)
M
(N) =
∑
α∈A
[(Cα)M (N)], (1.3)
(
∧
α∈A
Cα
)
M
(N) =
⋂
α∈A
[(Cα)M (N)], (1.4)
for every family {Cα ∈ CO(R) | α ∈ A} and every N ⊆ M . The class
CO(R) relative to these operations is a complete “big” lattice. In particular,
CO(R) possesses the greatest element R, where ( R)M (N) = M , as well
as the least element R, where ( R)M (N) = N for every N ⊆ M .
2. Mappings of closure operators in adjoint situation
Throughout of this paper we consider a pair of covariant adjoint
functors H = HomR(U, -) and T = U⊗R-, determined by the bimodule RUS
“adm-n1” — 2018/4/2 — 12:46 — page 100 — #102
100 Closure operators in modules and adjoint functors
(see Section 1). Now we will define two mappings which operate between
the classes of closure operators CO(R) and CO(S) of the categories R-Mod
and S-Mod. We essentially use some peculiarities of studied situation, in
particular, the natural transformations Φ and Ψ with the conditions (1.1)
and (1.2).
I. Mapping C 7→ C
∗ from CO(R) to CO(S)
Let C ∈ CO(R), Y ∈ S-Mod and n : N
⊆
−→ Y be an arbitrary inclusion
of S-Mod. We will construct a new function C∗ in S-Mod as follows.
Applying T we obtain the morphism T (n) : T (N) → T (Y ) of R-Mod.
Using the operator C, we have the following decomposition of T (n):
T (N)
T (n)
22
T (n) // ImT (n)
⊆ // CT (Y )
(
ImT (n)
) ⊆ // T (Y ),
where T (n) is the restriction of T (n) to its image. We consider the nat-
ural R-morphism πn
C : T (Y ) → T (Y )/CT (Y )
(
ImT (n)
)
. Applying H and
using ΨY , we obtain the composition of morphisms:
Y
ΨY−−−→ HT (Y )
H(πn
C)
−−−−−→ H[T (Y )/CT (Y )
(
ImT (n)
)
].
Definition 1. For every operator C ∈ CO(R) and every inclusion
n : N
⊆
−−→ Y of S-Mod, we define the function C∗ by the rule:
C∗
Y (N) = Ker[H(πn
C) ·ΨY ]. (2.1)
Proposition 2.1. The function C∗ defined by (2.1) is a closure operator
of the category S-Mod.
Proof. We will verify, for the function C∗, the conditions (c1)–(c3) of the
definition of closure operator (Section 1).
(c1) By Definition 1 ImT (n) ⊆ CT (Y )
(
ImT (n)
)
= Kerπn
C, so πn
C ·
T (n) = 0, therefore H(πn
C) ·HT (n) = 0. By the naturality of Ψ we have
ΨY · n = HT (n) ·ΨN , therefore
[H(πn
C) ·ΨY · n](N) = [H(πn
C) ·HT (n) ·ΨN ](N) = 0.
This means that N ⊆ Ker[H(πn
C) ·ΨY ] = C∗
Y (N), so (c1) is true.
(c2) Let N1, N2 ∈ L(Y ) and N1 ⊆ N2. We denote the existing in-
clusions as follows: i : N1
⊆
−−→ N2, n1 : N1
⊆
−−→ Y , n2 : N2
⊆
−−→ Y , so
n1 = n2 · i and therefore T (n1) = T (n2) ·T (i). Then ImT (n1) ⊆ ImT (n2)
“adm-n1” — 2018/4/2 — 12:46 — page 101 — #103
A. I. Kashu 101
and CT (Y )
(
ImT (n1)
)
⊆ CT (Y )
(
ImT (n2)
)
. This relation implies the mor-
phism π : T (Y )/CT (Y )
(
ImT (n)
)
→ T (Y )/CT (Y )
(
ImT (n2)
)
, which de-
fines the morphism H(π) of the following diagram in S-Mod:
H[T (Y )/CT (Y )(ImT (n1))]
H(π)
��
Y
ΨY // HT (Y )
H(π
n1
C )
44
H(π
n2
C )
**
H[T (Y )/CT (Y )(ImT (n2))].
Therefore Ker[H(πn1
C ) · ΨY ] ⊆ Ker[H(πn2
C ) · ΨY ], which means that
C∗
Y (N1) ⊆ C∗
Y (N2), so (c2) is true.
(c3) Let f : Y → Y ′ be an arbitrary S-morphism and n : N
⊆
−→ Y
be an inclusion. We denote n′ : f(N)
⊆
−−→ Y ′. Then the R-morphism
T (f) : T (Y ) → T (Y ′) implies the morphism (T (f))′ : ImT (n)→ ImT (n′),
as well as the morphism (T (f))′′ : CT (Y )
(
ImT (n)
)
→ CT (Y ′)
(
ImT (n′)
)
,
by which we obtain the morphism π : T (Y )/CT (Y )
(
ImT (n)
)
→
T (Y ′)/CT (Y ′)
(
ImT (n′)
)
. Then we have in S-Mod the diagram:
Y
ΨY //
f
��
HT (Y )
H(πn
C)
//
HT (f)
��
H[T (Y )/CT (Y )(ImT (n))]
H(π)
��
Y ′
Ψ
Y ′
// HT (Y ′)
H(πn′
C )
// H[T (Y ′)/CT (Y ′)(ImT (n′))],
where H(π) ·H(πn
C) ·ΨY = H(πn′
C ) ·ΨY ′ · f . Therefore:
f
(
Ker[H(πn
C) ·ΨY ]
)
⊆ Ker[H(πn′
C ) ·ΨY ′ ],
and by definition this means that f
(
C∗
Y (N)
)
⊆ C∗
Y ′
(
f(N)
)
, so (c3) is true,
which ends the proof.
II. Mapping D 7→ D
∗ from CO(S) to CO(R)
Now we will define in our adjoint situation (T,H) an inverse mapping
from CO(S) to CO(R). Let D ∈ CO(S) and m : M
⊆
−−→ X be an inclusion
of R-Mod. Then in S-Mod we have the morphism H(m) : H(M) → H(X)
“adm-n1” — 2018/4/2 — 12:46 — page 102 — #104
102 Closure operators in modules and adjoint functors
and by operator D we obtain the following decomposition of H(m):
H(M)
H(m)
&&H(m)
∼=
// ImH(m)
jm
D
⊆
// DH(X) (ImH(m))
im
D
⊆
// H(X)
(we remark that H(m) is a monomorphism, so its restriction H(m) is an
isomorphism).
Now using T and Φ we have in R-Mod the situation:
TH(M)
TH(m)
((
// T
[
DH(X)
(
ImH(m)
)] T (imD )
// TH(X)
ΦX // X.
Definition 2. For every closure operator D ∈ CO(S) and every inclusion
m : M
⊆
−−→ X of R-Mod we define the function D∗ by the rule:
D∗
X(M) = Im[ΦX · T (imD )] +M. (2.2)
Proposition 2.2. The function D∗ defined by (2.2) is a closure operator
of R-Mod.
Proof. (c1) By Definition 2 it is clear that M ⊆ D∗
X(M).
(c2) Let M1,M2 ∈ L(X) and κ : M1
⊆
−−→ M2. We denote m1 : M1
⊆
−→
X and m2 : M2
⊆
−−→ X, so m1 = m2 ·κ and H(m1) = H(m2) ·H(κ). Then
we have in S-Mod the following situation:
H(M1)
H(m1)
%%H(m1)
∼=
//
H(k)
��
ImH(m1)
j
m1
D
⊆
//
H(k)
��
DH(X)
(
ImH(m1)
)
D(H(k))
��
i
m1
D
⊆
// H(X)
H(M2)
H(m2)
44
H(m2)// ImH(m2)
j
m2
D
⊆
// DH(X)
(
ImH(m2)
) i
m2
D
⊆
// H(X).
“adm-n1” — 2018/4/2 — 12:46 — page 103 — #105
A. I. Kashu 103
Here the morphism H(κ) implies H(κ), as well as D
(
H(κ)
)
. Coming back
in R-Mod by T , we obtain the diagram:
TH(M1)
TH(m1)
''
TH(k)
��
// T
[
DH(X)
(
ImH(m1)
)]
T (i
m1
D )
((
T [D(H(k))]
��
TH(X)
ΦX // X.
TH(M2)
TH(m2)
99
// T
[
DH(X)
(
ImH(m2)
)]
T (i
m2
D )
66
We have T (im1
D ) = T (im2
D ) ·T [D(H(k))], therefore ImT (im1
D ) ⊆ ImT (im2
D ),
which shows that Im[ΦX · T (im1
D )] ⊆ Im[ΦX · T (im2
D )]. Adding M to both
parts, by definition we have D∗
X(M1) ⊆ D∗
X(M2), so (c2) is true.
(c3) Let f : X → X ′ be a morphism of R-Mod and m : M
⊆
−−→ X. We
will verify the relation: f
(
D∗
X(M)
)
⊆ D∗
X′
(
f(M)
)
. For that we denote:
m′ : f(M)
⊆
−−→ X ′ and f ′ : M → f(M) is the restriction of f , i.e. f ·m =
m′ · f ′. Applying H and using D, we obtain in S-Mod the situation:
H(M)
H(m)
&&H(m)
∼=
//
H(f ′)
��
ImH(m)
jmD
⊆
//
H(f)
��
DH(X)
(
ImH(m)
)
D(H(f))
��
imD
⊆
// H(X)
H(f)
��
H
(
f(M)
)
H(m′)
33
H(m′)
∼=
// ImH(m′)
jm
′
D
⊆
// DH(X′)
(
ImH(m′)
) im
′
D
⊆
// H(X ′),
where D
(
H(f)
)
is defined by the morphism H(f).
Using T and Φ, we obtain in R-Mod the diagram:
TH(M)
TH(m)
((
//
TH(f ′)
��
T
[
DH(X)
(
ImH(m)
)] T (imD )
//
T [D(H(f))]
��
TH(X)
TH(f)
��
ΦX // X
f
��
TH
(
f(M)
)
TH(m′)
44
// T
[
DH(X′)
(
ImH(m′)
)] T (im
′
D )
// TH(X ′)
Φ
X′
// X ′.
“adm-n1” — 2018/4/2 — 12:46 — page 104 — #106
104 Closure operators in modules and adjoint functors
We have f ·ΦX · T (imD ) = ΦX′ · T (im
′
D ) · T [D
(
H(f)
)
], therefore Im[f ·ΦX ·
T (imD )] ⊆ Im[ΦX′ ·T (im
′
D )], which implies f
(
Im[ΦX ·T (i
m
D )]+M
)
⊆ Im[ΦX′ ·
T (im
′
D )] + f(M). By definition this means that f
(
D∗
X(M)
)
⊆ D∗
X′
(
f(M)
)
,
i.e. (c3) is true, which ends the proof.
3. Particular cases
As examples in continuation we verify the effect of “star” mappings
defined above in some particular cases, namely for the extreme (trivial)
elements of the lattices of closure operators, i.e. C ∈ { R, R} ⊆ CO(R)
and D ∈ { S, S} ⊆ CO(S).
1. Let C = R, where R is the least element of CO(R), i.e.
( R)X(M) = M for every M ⊆ X. By construction of C∗, in this case
for every inclusion n : N
⊆
−→ Y of S-Mod we have such decomposition
of T (n):
T (N)
T (n)
33
T (n) // ImT (n) = CT (Y )
(
ImT (n)
) ⊆ // T (Y ).
By natural epimorphism πn
C : T (Y ) → T (Y )/ ImT (n) and applying H we
obtain in S-Mod the composition:
Y
ΨY−−−→ HT (Y )
H(πn
C)
−−−−−→ H[T (Y )/ ImT (n)].
By definition of C∗ we have C∗
Y (N) = Ker[H(πn
C) ·ΨY ]. We denote this
operator by D◦, so D◦
Y (N)
def
== Ker[H(πn
C) ·ΨY ]. Therefore it is verified
that ∗
R = D◦.
2. LetC = R, where R is the greatest element ofCO(R), i.e. CX(M) =
X for every M ⊆ X. For the inclusion n : N
⊆
−−→ Y of S-Mod we have in
R-Mod:
T (N)
T (n)
−−−−→ ImT (n)
⊆
−−→ CT (Y )
(
ImT (n)
)
= T (Y ),
so in S-Mod we obtain the composition:
Y
ΨY−−−→ HT (Y )
0
−−−→ H(0) = 0
(since πn
C = 0). Therefore Ker[0·ΨY ] = Ker 0 = Y and we have C∗
Y (N) = Y
for every N ⊆ Y , which means that ∗
R = S.
“adm-n1” — 2018/4/2 — 12:46 — page 105 — #107
A. I. Kashu 105
3. Let D = S, where S is the least element of CO(S), i.e. DY (N) = N
for every n : N
⊆
−−→ Y of S-Mod. Then for every inclusion m : M
⊆
−−→ X
of R-Mod we have in S-Mod the situation:
H(M)
H(m)
22
H(m)
∼=
// ImH(m) = DH(X)
(
ImH(m)
) imD
⊆
// H(X).
Now by T and Φ we obtain in S-Mod:
TH(M)
TH(m)
33
T (H(m))
∼=
// T
(
ImH(m)
)
= T
[
DH(X)
(
ImH(m)
)] T (imD ) // TH(X)
ΦX // X.
Since T (H(m)) is an isomorphism and using the naturality relation
ΦX · TH(m) = m · ΦM , we have:
Im[ΦX · T (imD )] = Im[ΦX · TH(m)] = Im[m · ΦM ] = ImΦM ⊆ M.
By definition now it is clear that:
D∗
X(M) = Im[ΦX · T (imD )] +M = M
for every M ⊆ X, i.e. D∗ = R or ∗
S = R.
4. Let D = S, where S is the greatest element of CO(S), i.e.
DY (N) = Y for every N ⊆ Y . Then for every inclusion m : M
⊆
−−→ X of
R-Mod we have in S-Mod the situation:
H(M)
H(m)
44
H(m)
∼=
// ImH(m)
jm
D
⊆
// DH(X)
(
ImH(m)
)
im
D
H(X).
By T and Φ we obtain in R-Mod:
TH(M)
TH(m)
33
ΦM
��
T (H(m))
∼=
// T
(
ImH(m)
)
T (jm
D
)
// T
[
DH(X)
(
ImH(m)
)]
))
T (im
D
)
TH(X)
ΦX
��
M
⊆
m // X.
Therefore in this case Im[ΦX ·T (imD )] = ImΦX and D∗
X(M) = ImΦX +M .
We denote this operator by C◦, i.e. C◦
X(M)
def
== ImΦX +M , so it is proved
that ∗
S = C◦.
“adm-n1” — 2018/4/2 — 12:46 — page 106 — #108
106 Closure operators in modules and adjoint functors
Totalizing the mentioned above facts, we can present the general
situation on images of extreme elements:
CO(R)
(−)∗ // CO(S)
(−)∗
oo
Proposition 3.1. The “star” mappings act on the extreme closure opera-
tors as follows:
∗
R = D◦, ∗
R = S;
∗
S = R,
∗
S = C◦.
4. Partial order and “star” mappings
In this section we will study the behaviour of the mappings C 7→ C∗
and D 7→ D∗ relative to the partial order in the classes CO(R) and CO(S).
Proposition 4.1. The “star” mappings are monotone, i.e. they preserve
the relations of partial order:
a) C1 6 C2 ⇒ C∗
1 6 C∗
2 ;
b) D1 6 D2 ⇒ D∗
1 6 D∗
2.
Proof. a) We verify the monotony of the mapping C 7→ C∗ from CO(R)
to CO(S). Let C1, C2 ∈ CO(R) and C1 6 C2. For every inclusion
n : N
⊆
−−→ Y of S-Mod by the construction of Definition 1 and using
the relation C1 6 C2 we have: (C1)T (Y )
(
ImT (n)
)
⊆ (C2)T (Y )
(
ImT (n)
)
.
This implies in R-Mod the morphism π from the diagram:
T (Y )/(C1)T (Y )
(
ImT (n)
)
π
��
T (Y )
πn
C1
55
πn
C2
))
T (Y )/(C2)T (Y )
(
ImT (n)
)
,
“adm-n1” — 2018/4/2 — 12:46 — page 107 — #109
A. I. Kashu 107
where πn
C1
and πn
C2
are the natural morphisms. By H and Ψ we obtain in
S-Mod the situation:
H
[
T (Y )/(C1)T (Y )
(
ImT (n)
)]
H(π)
��
Y
ΨY // HT (Y )
H(πn
C1
)
55
H(πn
C2
)
))
H
[
T (Y )/(C2)T (Y )
(
ImT (n)
)]
,
where H(π) ·H(πn
C1
) ·ΨY = H(πn
C1
) ·ΨY . Therefore
Ker[H(πn
C1
) ·ΨY ] ⊆ Ker[H(πn
C2
) ·ΨY ],
which by definition means that (C∗
1 )Y (N) ⊆ (C∗
2 )Y (N) for every N ⊆ Y ,
i.e. C∗
1 6 C∗
2 .
b) Now we will verify the monotony of the mapping D 7→ D∗ from
CO(S) to CO(R). Let D1, D2 ∈ CO(S) and D1 6 D2. For an arbitrary
inclusion m : M
⊆
−−→ X of R-Mod we follow the construction of operators
D∗
1 and D∗
2. Since D1 6 D2, we have the inclusion i of the diagram:
(D1)H(X)
(
ImH(m)
)
imD1
⊆ ((
⊇ i
��
H(M)
H(m)
∼=
// ImH(m)
jmD1
⊆
66
jmD2
⊆ ((
H(X).
(D2)H(X)
(
ImH(m)
)
imD2
⊆
66
Therefore in R-Mod we obtain the situation:
T [(D1)H(X)(ImH(m))]
T (imD1
)
((
T (i)
��
TH(M)
T (H(m))
∼=
// T (ImH(m))
T (jmD1
)
55
T (jmD2
)
))
TH(X)
ΦX // X.
T [(D2)H(X)(ImH(m))]
T (imD2
)
66
By commutativity of diagram we have Im[ΦX ·T (i
m
D1
)] ⊆ Im[ΦX ·T (i
m
D2
)]
and adding M to both parts by definition we obtain that (D1)
∗
X(M) ⊆
(D2)
∗
X(M) for every M ⊆ X, i.e. D∗
1 6 D∗
2.
“adm-n1” — 2018/4/2 — 12:46 — page 108 — #110
108 Closure operators in modules and adjoint functors
We remark that from the particular cases of Section 3 and by monotony
of “star” mappings follows
Corollary 4.2. a) For every operator C ∈ CO(R) we have C∗ > D◦.
b) For every operator D ∈ CO(S) we have D∗ 6 C◦. �
In continuation we will prove some more properties of “star” mappings,
related to the partial order in CO(R) and CO(S).
Proposition 4.3. a) For every operator C ∈ CO(R), the relation C>C∗∗
is true.
b) For every operator D ∈ CO(S), the relation D 6 D∗∗ is true.
Proof. a) Let C ∈ CO(R) and m : M
⊆
−−→ X be an arbitrary inclusion of
R-Mod. Then in S-Mod we have the morphism H(m) : H(M) → H(X).
We follow the construction of C∗ for the inclusion n : ImH(m)
⊆
−−→ H(X).
In S-Mod we have:
H(M)
H(m)
33
H(m)
∼=
// ImH(m)
n
⊆
// H(X).
Using T and C we obtain in R-Mod:
TH(M)
TH(m)
22
TH(m)
��
T (H(m)) // T
(
ImH(m)
)
T (n)
��
T (n) // TH(X)
ImTH(m) ImT (n)
⊆
// CTH(X)
(
ImT (n)
)
.
⊆
OO
Now we consider the natural morphism
πn
C : TH(X) → TH(X)/CTH(X)
(
ImT (n)
)
.
Applying H and adding ΨH(X), we have in S-Mod:
H(X)
ΨH(X)
−−−−−→ HTH(X)
H(πn
C)
−−−−−−→ H[TH(X)/CTH(X)
(
ImT (n)
)
].
By Definition 1 we have:
C∗
H(X)
(
ImH(m)
)
= Ker[H(πn
C) ·ΨH(X)]. (4.1)
“adm-n1” — 2018/4/2 — 12:46 — page 109 — #111
A. I. Kashu 109
We denote inC∗ : C∗
H(X)
(
ImH(m)
) ⊆
−−→ H(X) and consider the following
commutative diagram in R-Mod:
TH(X)
1TH(X)
))T (ΨH(X)) // THTH(X)
ΦTH(X) //
TH(πn
C)
��
TH(X)
ΦX //
πn
C
��
X
πm
C
��
T
[
C∗
H(X)
(
ImH(m)
)] 0 //
T (in
C∗ )
OO
TH[TH(X)/A]
Φ
TH(X)// TH(X)/A
(1/C)ΦX// X/CX(M),
where A = CTH(X)
(
ImT (n)
)
and (1/C)ΦX is defined by ΦX . From the
definition of C∗
H(X)
(
ImH(m)
)
(see (4.1)) we have H(πn
C) ·ΨH(X) · i
n
C∗ = 0,
therefore TH(πn
C) ·T (ΨH(X)) ·T (i
n
C∗) = 0. From commutativity of diagram
we obtain πm
C ·ΦX ·1TH(X)·T (i
n
C∗) = 0, so Im[ΦX ·T (i
n
C∗)] ⊆ Kerπn
C = CX(M).
Since M ⊆ CX(M), now we have Im[ΦX · T (in
C∗)] +M ⊆ CX(M). The left
part of this relation by definition represents the module C∗∗
X (M), therefore
we obtain C∗∗
X (M) ⊆ CX(M), for every M ⊆ X, i.e. C∗∗ 6 C proving a).
b) To verify the part b) we consider an operator D ∈ CO(S) and an
inclusion n : N
⊆
−−→ Y of S-Mod. Using the operator D∗ ∈ CO(R) we
obtain the decomposition of T (n):
T (N)
T (n)
22
T (n) // ImT (n)
m
⊆
))jn
D∗
⊆
// D∗
T (Y )
(
ImT (n)
)
in
D∗
⊆
// T (Y ).
We denote by m the inclusion m : ImT (n)
⊆
−−→ T (Y ) and by πm
D∗ the
natural morphism πm
D∗ : T (Y ) → T (Y )/D∗
T (Y )
(
ImT (n)
)
. Applying H and
using Ψ, we obtain in S-Mod the composition:
Y
ΨY−−−→ HT (Y )
H(πm
D∗ )
−−−−−−−→ H
[
T (Y )/D∗
T (Y )
(
ImT (n)
)]
and by definition we have:
D∗∗
Y (N) = Ker[H(πm
D∗) ·ΨY ]. (4.2)
Now we apply the transition D 7→ D∗ to the inclusion m : ImT (n)
⊆
−−→
T (Y ) of R-Mod. With the help of H we have in S-Mod the situation:
HT (N)
HT (n)
44
H(T (n))// H
(
ImT (n)
)
H(m)
((H(m)
∼=
// ImH(m)
jm
D
⊆
// DHT (Y )
(
ImH(m)
)
im
D
⊆
//HT (Y ).
“adm-n1” — 2018/4/2 — 12:46 — page 110 — #112
110 Closure operators in modules and adjoint functors
Returning in R-Mod and using Φ, we obtain the diagram:
THT (N)
THT (n)
22TH(T (n))//
ΦT (N)
��
TH(ImT (m))
TH(m)
((T (H(m))//
ΦImT (n)
��
T (ImH(m))
T (jm
D
)
// T [DHT (Y )(ImH(m))]
T (im
D
)
//
Φm
T (Y )
''
Φm
T (Y)
��
THT (Y )
ΦT (Y )
��
T (N) // ImT (n)
jn
D∗
⊆
// D∗
T (Y )(ImT (n))
in
D∗
⊆
// T (Y ),
where Φm
T (Y ) = ΦT (Y ) · T (i
m
D ). By Definition 2 we have:
D∗
T (Y )
(
ImT (n)
)
= Im[ΦT (Y ) · T (i
m
D )] + ImT (n). (4.3)
We denote by im
D∗ the inclusion im
D∗ : D∗
T (Y )
(
ImT (n)
) ⊆
−−→ T (Y ) and
by πm
D∗ the natural morphism πm
D∗ : T (Y ) → T (Y )/D∗
T (Y )
(
ImT (n)
)
, so
πm
D∗ · imD∗ = 0.
Using D and Ψ, we obtain in S-Mod the diagram:
HT (N)
H
(
T (n)
)
��
N
ΨNoo
⊆
//
Ψ′
N
��
DY (N)
Ψ′′
N
��
k
⊆
// Y
ΨY
��
H
(
ImT (n)
) H(m) // ImH(m)
⊆
// DHT (Y )
(
ImH(m)
)
⊆
// HT (Y ),
where Ψ′
N = H(m) ·H
(
T (n)
)
·ΨN and κ : DY (N)
⊆
−−→ Y is the inclusion.
The morphism ΨY implies the morphism Ψ′′
N , by which (using the last
but one diagram) we obtain in S-Mod:
DY (N)
Ψ′′
N //
k⊇
��
DHT (Y )(ImH(m))
im
D⊇
��
ΨDHT (Y )
(ImH(m))
// HT [DHT (Y )(ImH(m))]
H(Φm
T (Y )
)
// H[D∗
T (Y )(ImT (n))]
H(im
D∗ )
uu
0
��
Y
ΨY //HT (Y )
1HT (Y ) //HT (Y )
H(πm
D∗ )
//H[T (Y )/D∗
T (Y )(ImT (n))],
where 1HT (Y ) = H(ΦT (Y ))·ΨHT (Y ). As we mentioned above, by construction
πm
D∗ · im
D∗ = 0, therefore H(πm
D∗) ·H (im
D∗) = 0. Therefore:
H(πm
D∗) · 1TH(Y ) ·ΨY · k
= H(πm
D∗) ·H (imD∗) ·H
(
Φm
T (Y )
)
·ΨDHT (Y )(ImH(m)) ·Ψ
′′ = 0.
This shows that DY (N) ⊆ Ker[H(πm
D∗) ·ΨY ]
def
== D∗∗
Y (N) for every N ⊆ Y ,
which means that D 6 D∗∗.
“adm-n1” — 2018/4/2 — 12:46 — page 111 — #113
A. I. Kashu 111
Remark. In this case we mention that the proved above facts are perfectly
concordant with the results for preradicals in adjoint situation, where
r > r∗∗ and s 6 s∗∗ for every preradicals r of R-Mod and s of S-Mod
([5, 6]).
5. Lattice operations and “star” mappings
Now we will study the behaviour of “star” mappings in the adjoint
situation (T,H) with respect to lattice operations “∧” (meet) and “∨”
(join) in the classes CO(R) and CO(S).
Proposition 5.1. The mapping C 7→ C∗ from CO(R) to CO(S) preserves
the meet of closure operators, i.e.
(
∧
α∈A
Cα
)∗
=
∧
α∈A
(
Cα
)∗
for every family of operators {Cα ∈ CO(R) | α ∈ A}.
Proof. Let {Cα ∈ CO(R) | α ∈ A} be an arbitrary family of closure
operators of R-Mod and n : N
⊆
−−→ Y be an inclusion of S-Mod. By
definition of mapping C 7→ C∗, for any α ∈ A we have in R-Mod the
morphisms:
T (N)
T (n)
''T (n)// ImT (n)
jn
Cα
⊆
// (Cα)
T (Y )
(ImT (n))
in
Cα
⊆
// T (Y )
πn
Cα // T (Y )/(Cα)
T (Y )
(ImT (n)).
Using H and Ψ, we obtain in S-Mod the composition:
Y
ΨY−−−→ HT (Y )
H(πn
Cα
)
−−−−−−−→ H
[
T (Y )/(Cα)T (Y )
(
ImT (n)
)]
and by Definition 1 we have: (Cα)
∗
Y (N) = Ker[H(πn
Cα
) ·ΨY ].
Similarly, for C =
∧
α∈A
Cα from the definition of C∗ we have:
(
∧
α∈A
Cα
)∗
Y
(N) = Ker[H(πn
∧Cα
) ·ΨY ],
where πn
∧Cα
: T (Y ) → T (Y )/
(
∧
α∈A
Cα
)
T (Y )
(
ImT (n)
)
is the natural morphism.
Now we observe that is true the equality:
Ker[H(πn
∧Cα
)] =
⋂
α∈A
[Ker[H(πn
Cα
)], (5.1)
“adm-n1” — 2018/4/2 — 12:46 — page 112 — #114
112 Closure operators in modules and adjoint functors
since Ker(πn
∧Cα
) =
(
∧
α∈A
Cα
)
T (Y )
(
ImT (n)
)
=
⋂
α∈A
[(
Cα
)
T (Y )
(
ImT (n)
)]
=
⋂
α∈A
[Ker(πn
Cα
)]. Using this relation we obtain:
(
∧
α∈A
Cα
)∗
Y
(N)
def
== Ker[H(πn
∧Cα
) ·ΨY ] = Ψ−1
Y
[
KerH(πn
∧Cα
)
]
(5.1)
== Ψ−1
Y [
⋂
α∈A
KerH(πn
Cα
)] =
⋂
α∈A
[Ψ−1
Y
(
KerH(πn
Cα
)
)
]
=
⋂
α∈A
Ker[H(πn
Cα
) ·ΨY ] =
⋂
α∈A
[(
Cα
)∗
Y
(N)
]
=
(
∧
α∈A
C∗
α
)
Y
(N)
for every N ⊆ Y . This shows that
(
∧
α∈A
Cα
)∗
=
∧
α∈A
(Cα)
∗.
Proposition 5.2. The mapping D 7→ D∗ from CO(S) to CO(R) pre-
serves the join of closure operators, i.e.
(
∨
α∈A
Dα
)∗
=
∨
α∈A
(Dα)
∗
for every family of operators {Dα ∈ CO(S) | α ∈ A}.
Proof. Let {Dα ∈ CO(S) | α ∈ A} be an arbitrary family of
closure operators of S-Mod and m : M
⊆
−−→ X be an inclusion
of R-Mod. By definition (Dα)
∗
X(M) = Im[ΦX · T (imDα
)] + M , where
imDα
: (Dα)H(X)
(
ImH(m)
) ⊆
−−→ H(X). The same rule applied for the
operator
∨
α∈A
Dα and inclusion m leads to equality:
(
∨
α∈A
Dα
)∗
X
(M) = Im[ΦX · T (im∨Dα
)] +M,
where im∨Dα
:
(
∨
α∈A
Dα
)
H(X)
(
ImH(m)
) ⊆
−−→ H(X). Since
Im
(
im∨Dα
)
=
∑
α∈A
Im
(
imDα
)
,
we obtain:
ImT (im∨Dα
) =
∑
α∈A
ImT
(
imDα
)
. (5.2)
“adm-n1” — 2018/4/2 — 12:46 — page 113 — #115
A. I. Kashu 113
Using this equality, by definitions we have:
(
∨
α∈A
Dα
)∗
X
(M) = Im[ΦX · T (im∨Dα
)] +M = ΦX [ImT (im∨Dα
)] +M
(5.2)
== ΦX
[
∑
α∈A
ImT
(
imDα
)]
+M =
[
∑
α∈A
ΦX
(
ImT (imDα
)
)]
+M
=
(
∑
α∈A
Im[ΦX · T (imDα
)]
)
+M =
∑
α∈A
(
Im[ΦX · T (imDα
)] +M
)
=
∑
α∈A
[(
Dα
)∗
X
(M)
]
=
(
∨
α∈A
D∗
α
)
X
(M)
for every M ⊆ X. Therefore we obtain
(
∨
α∈A
Dα
)∗
=
∨
α∈A
(
Dα
)∗
.
6. Product of closure operators and “star” mappings
We remember that besides lattice operations, in the class of closure
operators CO(R) also the operation of multiplication is defined by the
rule:
(C1 · C2)X(M) = (C1)X [(C2)X(M)]
for every operators C1, C2 ∈ CO(R) and M ⊆ X. In continuation we will
show how the “star” mappings act to the product of closure operators.
Proposition 6.1. For every closure operators C1, C2 ∈ CO(R) the rela-
tion (C1 · C2)
∗ > C∗
1 · C∗
2 is true.
Proof. Let C1, C2 ∈ CO(R) and n : N
⊆
−−→ Y be an arbitrary inclusion of
S-Mod. By definitions we have:
(C1 · C2)
∗
Y (N) = Ker[H(πn
C1·C2
) ·ΨY ],
where πn
C1·C2
: T (Y ) → T (Y )/(C1 · C2)T (Y )
(
ImT (n)
)
is the natural mor-
phism.
On the other hand, to define [C∗
1 · C∗
2 ]Y (N) we consider in S-Mod the
inclusions:
N
n
55
l // (C2)
∗
Y
(N)
κ // Y,
i.e. n = κ · l. Therefore T (n) = T (κ) · T (l) and ImT (n) ⊆ ImT (κ).
Now we apply the transition C1 7→ C∗
1 for the inclusion κ:
ImT (κ)
⊆
−→ (C1)T (Y )
(
ImT (κ)
) ⊆
−→ T (Y )
πκ
C1−−→ T (Y )/(C1)T (Y )
(
ImT (κ)
)
.
“adm-n1” — 2018/4/2 — 12:46 — page 114 — #116
114 Closure operators in modules and adjoint functors
By Definition 1 we have:
(C∗
1 · C∗
2 )Y (N) = (C1)
∗
Y [(C2)
∗
Y (N)] = Ker[H(πκ
C1
) ·ΨY ].
Similarly
(C2)
∗
Y (N) = Ker[H(πn
C2
) ·ΨY ],
where πn
C2
: T (Y ) → T (Y )/(C2)T (Y )
(
ImT (n)
)
is the natural morphism.
So in S-Mod we obtain the situation:
(C2)
∗
Y
(N)
k
−−→
⊆
Y
ΨY−−→ HT (Y )
H(πn
C2
)
−−−−−→ H[T (Y )/(C2)T (Y )
(
ImT (n)
)
],
where by construction H(πn
C2
) ·ΨY · κ = 0.
Applying T and completing the diagram we have in R-Mod:
T [(C2)
∗
Y
(N)]
T (κ) // T (Y )
πn
C2 ''
T (ΨY ) // THT (Y )
ΦTY
oo
TH(πn
C2
)
// TH[T (Y )/(C2)T (Y )
(ImT (n))]
Φ
T (Y )tt
T (Y )/(C2)T (Y )
(ImT (n)).
By naturality of Φ the equality πn
C2
· ΦT (Y ) = ΦT (Y ) · TH(πn
C2
) is true,
therefore
Φ T (Y ) · TH(πn
C2
) · T (ΨY ) = πn
C2
· ΦT (Y ) · T (ΨY ) = πn
C2
· 1T (Y ) = πn
C2
.
From the remark that H(πn
C2
) ·ΨY ·κ = 0 it follows that TH(πn
C2
) ·T (ΨY ) ·
T (κ) = 0. Therefore
ImT (κ) ⊆ Ker[TH(πn
C2
)× T (ΨY )]
⊆ Ker[Φ T (Y ) · TH(πn
C2
) · T (ΨY )]= Kerπn
C2
=(C2)T (Y )
(
ImT (n)
)
,
i.e. ImT (κ) ⊆ (C2)T (Y )
(
ImT (n)
)
. This relation implies the inclusion:
(C1)T (Y )
(
ImT (κ)
)
⊆ (C1)T (Y )
[
(C2)T (Y )
(
ImT (n)
)]
def
== (C1 · C2)T (Y )
(
ImT (n)
)
,
which in its turn defines the epimorphism:
π : T (Y )/(C1)T (Y )
(
ImT (κ)
)
→ T (Y )/(C1 · C2)T (Y )
(
ImT (n)
)
.
“adm-n1” — 2018/4/2 — 12:46 — page 115 — #117
A. I. Kashu 115
Applying H we obtain in S-Mod the situation:
H[T (Y )/(C1)T (Y )(ImT (κ))]
H(π)
��
Y
ΨY // HT (Y )
H(πκ
C1
)
44
H(πn
C1·C2
)
**
H[T (Y )/(C1 · C2)T (Y )(ImT (n))].
Now it is obvious that Ker[H(πκ
C1
) ·ΨY ] ⊆ Ker[H(πn
C1·C2
) ·ΨY ], which by
definition means that (C∗
1 )Y [(C
∗
2 )Y (N)] ⊆ (C1 ·C2)
∗
Y (N) for every N ⊆ Y .
Therefore C∗
1 · C∗
2 6 (C1 · C2)
∗.
Similar statement takes place for the mapping D 7→ D∗.
Proposition 6.2. For every closure operators D1, D2 ∈ CO(S) the rela-
tion (D1 ·D2)
∗ 6 D∗
1 ·D
∗
2 is true.
Proof. Let D1, D2 ∈ CO(S) and m : M
⊆
−−→ X be an arbitrary inclusion
of R-Mod. We apply the mapping CO(S)
(−)∗
−−−−−−−→ CO(R) of Definition 2
in the following three cases.
1) For the product D1 ·D2 and inclusion m:
(D1 ·D2)
∗
X(M) = Im[ΦX · T (imD1·D2
)] +M,
where imD1·D2
: (D1 ·D2)H(X)
(
ImH(m)
) ⊆
−−→ H(X).
2) For the operator D2 and inclusion m:
(D2)
∗
X(M) = Im[ΦX · T (imD2
)] +M,
where imD2
: (D2)H(X)
(
ImH(m)
) ⊆
−−→ H(X).
3) For the operator D1 and inclusion κ : (D2)
∗
X(M)
⊆
−−→ H(X):
(D1)
∗
X [(D2)
∗
X(M)] = Im[ΦX · T (iκD1
)] +M,
where iκD1
: (D1)H(X)
(
ImH(κ)
) ⊆
−−→ H(X).
From the definition of (D2)
∗
X(M) we have in R-Mod the situation:
T
[(
D2
)
H(X)
(
ImH(m)
)]
T (imD2
)
//
f
,,
TH(X)
ΦX // X
(D2)
∗
X(M) = Im[ΦX · T
(
imD2
)
] +M,
⊆ κ
OO
“adm-n1” — 2018/4/2 — 12:46 — page 116 — #118
116 Closure operators in modules and adjoint functors
where f is the restriction of ΦX ·T (i
m
D2
) to (D2)
∗
X(M), i.e. κ·f = ΦX ·T (i
m
D2
).
Applying T we obtain in R-Mod the diagram:
HT
[(
D2
)
H(X)
(
ImH(m)
)]
HT (imD2
)
//
H(f)
**
HTH(X)
H(ΦX ) // H(X)
(
D2
)
H(X)
(
ImH(m)
)
imD2
⊆
//
Ψ(D2)H(X)(ImH(m))
OO
H(X)
1H(X)
::
ΨH(X)
OO
H[(D2)
∗
X(M)].
H(κ)
oo
H(κ)
OO
From its commutativity it follows that:
H(ΦX)·HT (imD2
)·Ψ(D2)H(X)(ImH(m))= H(ΦX)·ΨH(X)·i
m
D2
= 1H(X)· i
m
D2
= imD2
.
Therefore Im imD2
⊆ Im[H(ΦX) ·HT (imD2
)] = Im[H(κ) ·H(f)] ⊆ ImH(κ),
i.e.
(
D2
)
H(X)
(
ImH(m)
)
⊆ ImH(κ). This relation implies the inclusion:
(D1 ·D2)H(X) (ImH(m))
def
==
(
D1
)
H(X)
[(
D2
)
H(X)
(
ImH(m)
)] i
⊆
(
D1
)
H(X)
(
ImH(κ)
)
,
so in S-Mod we have the situation:
(
D1 ·D2
)
H(X)
(
ImH(m)
)
imD1·D2
⊆ ((
i ⊇
��
H(X),
(
D1
)
H(X)
(
ImH(κ)
)
iκD1
⊆
66
which implies in R-Mod the diagram:
T
[(
D1 ·D2
)
H(X)
(
ImH(m)
)]
T (imD1·D2
)
))
T (i)
��
TH(X)
ΦX // X.
T
[(
D1
)
H(X)
(
ImH(κ)
)]
T (iκD1
)
55
“adm-n1” — 2018/4/2 — 12:46 — page 117 — #119
A. I. Kashu 117
Now it is clear that Im[ΦX · T (imD1·D2
)] ⊆ Im[ΦX · T (iκD1
)]. Adding M to
both parts, by definition we have (D1 ·D2)
∗
X(M) ⊆ (D∗
1 ·D
∗
2)X(M) for
every X ⊆ M , therefore (D1 ·D2)
∗ 6 D∗
1 ·D
∗
2.
References
[1] L. Bican, P. Jambor, T. Kepka , P. Nemec, Preradicals and change of rings,
Comment. Math. Carolinae, 16, No.2, 1975, pp. 201–217.
[2] A. I. Kashu, Preradicals in adjoint situation, Mat. Issled., vyp. 48, 1978, pp. 48–64
(in Russian).
[3] A. I. Kashu, On correspondence of preradicals and torsions in adjoint situation,
Mat. Issled., vyp. 56, 1980, pp. 62–84 (in Russian).
[4] A. I. Kashu, Radicals and torsions in modules, Kishinev, Ştiinţa, 1983 (in Russian).
[5] A. I. Kashu, Radicals of modules and adjoint functors (preprint), Academy of
Sciences of MSSR, Institute of Mathematics. Kishinev, 1984 (in Russian).
[6] A. I. Kashu, Functors and torsions in categories of modules, Academy of Sciences
of RM, Institute of Mathematics. Kishinev, 1997 (in Russian).
[7] D. Dikranjan, E. Giuli, Factorizations, injectivity and compactness in categories
of modules, Commun. in Algebra, v. 19, No.1, 1991, pp. 45–83.
[8] D. Dikranjan, E. Giuli, Closure operators I, Topology and its applications, v. 27,
1987, pp. 129–143.
[9] D. Dikranjan, W. Tholen, Categorical structure of closure operators, Kluwer Aca-
demic Publishers, 1995.
[10] A. I. Kashu, Closure operators in the categories of modules, Part I, Algebra and
Discrete Math., v. 15 (2013), No.2, pp. 213–228.
[11] A. I. Kashu, Closure operators in the categories of modules, Part II, Algebra and
Discrete Math., v. 16 (2013), No.1, pp. 81–95.
[12] A. I. Kashu, Closure operators in the categories of modules,Part III, Bulet. Acad.
Şt. RM, Matematica, No.1(74), 2014, pp. 90–100.
[13] A. I. Kashu, Closure operators in the categories of modules, Part IV, Bulet. Acad.
Şt. RM, Matematica, No.3(76), 2014, pp. 13–22.
Contact information
A. I. Kashu Institute of Mathematics and Computer
Science, Academy of Sciences of Moldova,
5 Academiei str., Chişinău,
MD – 2028 MOLDOVA
E-Mail(s): alexei.kashu@math.md
Received by the editors: 07.07.2017.
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