Preradicals, closure operators in R-Mod and connection between them
For a module category R-Mod the class PR of preradicals and the class CO of closure operators are studied. The relations between these classes are realized by three mappings: Φ : CO → PR and Ψ₁, Ψ₂ : PR → CO. The impact of these mappings on the operations in PR and CO (meet, join, product, coproduct...
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irk-123456789-1533482019-06-15T01:28:01Z Preradicals, closure operators in R-Mod and connection between them Kashu, A.I. For a module category R-Mod the class PR of preradicals and the class CO of closure operators are studied. The relations between these classes are realized by three mappings: Φ : CO → PR and Ψ₁, Ψ₂ : PR → CO. The impact of these mappings on the operations in PR and CO (meet, join, product, coproduct) is investigated. It is established that in most cases the considered mappings preserve the lattice operations (meet and join), while the other two operations are converted one into another (i.e. the product into the coproduct and vice versa). 2014 Article Preradicals, closure operators in R-Mod and connection between them / A.I. Kashu // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 86–96. — Бібліогр.: 6 назв. — англ. 1726-3255 2010 MSC:16D90, 16S90, 06B23. http://dspace.nbuv.gov.ua/handle/123456789/153348 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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For a module category R-Mod the class PR of preradicals and the class CO of closure operators are studied. The relations between these classes are realized by three mappings: Φ : CO → PR and Ψ₁, Ψ₂ : PR → CO. The impact of these mappings on the operations in PR and CO (meet, join, product, coproduct) is investigated. It is established that in most cases the considered mappings preserve the lattice operations (meet and join), while the other two operations are converted one into another (i.e. the product into the coproduct and vice versa). |
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Article |
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Kashu, A.I. |
| spellingShingle |
Kashu, A.I. Preradicals, closure operators in R-Mod and connection between them Algebra and Discrete Mathematics |
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Kashu, A.I. |
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Kashu, A.I. |
| title |
Preradicals, closure operators in R-Mod and connection between them |
| title_short |
Preradicals, closure operators in R-Mod and connection between them |
| title_full |
Preradicals, closure operators in R-Mod and connection between them |
| title_fullStr |
Preradicals, closure operators in R-Mod and connection between them |
| title_full_unstemmed |
Preradicals, closure operators in R-Mod and connection between them |
| title_sort |
preradicals, closure operators in r-mod and connection between them |
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Інститут прикладної математики і механіки НАН України |
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2014 |
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http://dspace.nbuv.gov.ua/handle/123456789/153348 |
| citation_txt |
Preradicals, closure operators in R-Mod and connection between them / A.I. Kashu // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 86–96. — Бібліогр.: 6 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT kashuai preradicalsclosureoperatorsinrmodandconnectionbetweenthem |
| first_indexed |
2025-07-14T04:34:06Z |
| last_indexed |
2025-07-14T04:34:06Z |
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1837595527395409920 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 18 (2014). Number 1, pp. 86 – 96
© Journal “Algebra and Discrete Mathematics”
Preradicals, closure operators in R-Mod
and connection between them
A. I. Kashu
Abstract. For a module category R-Mod the class PR
of preradicals and the class CO of closure operators are studied.
The relations between these classes are realized by three mappings:
Φ : CO → PR and Ψ1, Ψ2 : PR → CO. The impact of these
mappings on the operations in PR and CO (meet, join, product,
coproduct) is investigated. It is established that in most cases the
considered mappings preserve the lattice operations (meet and join),
while the other two operations are converted one into another (i.e.
the product into the coproduct and vice versa).
0. Introduction and preliminary notions
In this work the preradicals and closure operators of a module category
R-Mod are studied ([1–6]). Three known mappings Φ : CO → PR, Ψ1, Ψ2 :
PR → CO realize the connection between the class PR of preradicals and
the class CO of closure operators of R-Mod ([1]). We study the influence
of these mappings on the operations in PR and CO: meet, join, product
and coproduct ([1, 2]).
At first we remind some necessary notions. Let R be a ring with unity
and R-Mod be the category of unitary left R-modules. For every module
M ∈ R-Mod we denote by L(M) the lattice of submodules of M .
A preradical r of R-Mod is a subfunctor of identity functor of
R-Mod, i.e. r(M) ⊆ M and f
(
r(M)
)
⊆ r(M ′) for every R-morphism
f : M → M ′ ([3, 4]).
2010 MSC: 16D90, 16S90, 06B23.
Key words and phrases: ring, module, lattice, preradical, closure operator,
product (coproduct) of closure operators.
A. I. Kashu 87
Let PR be the class of all preradicals of R-Mod. In PR four operations
are defined ([3]):
– the meet
∧
α∈A
rα of the family {rα|α ∈ A} ⊆ PR:
(
∧
α∈A
rα
)
(M) =
⋂
α∈A
rα(M);
– the join
∨
α∈A
rα of the family {rα|α ∈ A} ⊆ PR:
(
∨
α∈A
rα
)
(M) =
∑
α∈A
rα(M);
– the product r · s of preradicals r, s ∈ PR:
(r · s) (M) = r
(
s(M)
)
;
– the coproduct r : s of preradicals r, s ∈ PR:
[(r : s) (M)] / s(M) = r
(
M/ s(M)
)
,
for every module M ∈ R-Mod.
In PR the order relation “6 ” is defined by the rule:
r 6 s ⇔ r(M) ⊆ s(M) for every M ∈ R-Mod.
The class PR is a “big” complete lattice with respect to the operations
“ ∧” and “∨ ”.
A closure operator of R-Mod is a function C which associates to
every pair N ⊆ M , where N ∈ L(M), a submodule CM(N) ⊆ M with
the conditions:
(c1) N ⊆ CM(N);
(c2) if N1 ⊆ N2, then CM(N1) ⊆ CM(N2);
(c3) for every R-morphism f : M → M ′ and N ∈ L(M) we have
f
(
CM(N)
)
⊆ CM′
(
f(N)
)
.
We denote by CO the class of all closure operators of R-Mod. In this
class also four operations are defined ([1]):
– the meet
∧
α∈A
Cα of the family {Cα|α ∈ A} ⊆ CO:
(
∧
α∈A
Cα
)
M
(N) =
⋂
α∈A
[(Cα)M (N)];
88 Preradicals, closure operators in R-Mod
– the join
∨
α∈A
Cα of the family {Cα|α ∈ A} ⊆ CO:
(
∨
α∈A
Cα
)
M
(N) =
∑
α∈A
[(Cα)M (N)];
– the product C · D of the operators C, D ∈ CO:
(C · D)M(N) = CM
(
DM(N)
)
;
– the coproduct C # D of the operators C, D ∈ CO:
(C # D)M(N) = CD
M
(N)(N),
for every N ⊆ M .
In CO the order relation “6 ” is defined as follows:
C 6 D ⇔ CM(N) ⊆ DM(N) for every N ⊆ M.
The class CO is a “big” complete lattice with respect to the operations
“ ∧” and “∨ ”.
The connection between the classes CO and PR for a fixed module
category R-Mod is realized by the following three mappings ([1, 4]):
1) Φ : CO → PR, where we denote Φ(C) = rC for every C ∈ CO and
define:
rC(M) = CM(0)
for every M ∈ R-Mod;
2) Ψ1 : PR → CO, where Ψ1(r) = C r for every r ∈ PR and
[(C r)M(N)] / N = r(M/ N)
for every N ⊆ M ;
3) Ψ2 : PR → CO, where Ψ2(r) = Cr for every r ∈ PR and
(Cr)M(N) = N + r(M)
for every N ⊆ M .
It is well known that C r is the greatest among the operators C ∈ CO
with the property Φ(C) = r and, dually, Cr is the least among the
operators C ∈ CO for which Φ(C) = r. So, every preradical r ∈ PR gives
the equivalence class [Cr, C r] in CO, and every operator C ∈ CO defines
A. I. Kashu 89
the class [C r
C
, C r
C ] in which it is contained. The closure operators of the
form C r are called maximal. An operator C ∈ CO is maximal if and only if
CM(N) / N = CM/N(0̄) for every N ⊆ M
(
or: CM(N) / K = CM/K(N/ K)
for every K ⊆ N ⊆ M [6]
)
.
We denote by Max (CO) the class of maximal closure operators of
R-Mod. The mappings Φ and Ψ1 define a monotone bijection Max (CO) ∼=
PR. Dually, the closure operators of the form Cr are called minimal and
the mappings Φ and Ψ2 define a monotone bijection Min (CO) ∼= PR.
We remind also that the preradical r ∈ PR is called cohereditary if
r(M/ N) =
(
r(M) + N
)
/ N for every N ⊆ M ([3, 4]).
1. The mapping Φ and its impact on the operations of CO
We begin with the mapping Φ : CO → PR, where Φ(C) = rC and
rC(M) = CM(0) for every C ∈ CO and M ∈ R-Mod. We will study the
behaviour of this mapping relative to the operations in CO ([1, 2]).
Proposition 1.1. The mapping Φ preserves the operation of meet, i.e.
Φ
(
∧
α∈A
Cα
)
=
∧
α∈A
[
Φ (Cα)
]
for every family of operators {Cα|α ∈ A} ⊆ CO.
Proof. For every M ∈ R-Mod from the definitions it follows:
(
r∧
α∈A
Cα
)
(M) =
(
∧
α∈A
Cα
)
M
(0) =
⋂
α∈A
[(
Cα
)
M
(0)
]
=
=
⋂
α∈A
[
rCα(M)] =
(
∧
α∈A
rCα
)
(M).
Thus r∧
α∈A
Cα
=
∧
α∈A
rCα , so by our notations this proves the proposition.
Proposition 1.2. The mapping Φ preserves the operation of join, i.e.
Φ
(
∨
α∈A
Cα
)
=
∨
α∈A
[
Φ (Cα)
]
for every family of operators {Cα|α ∈ A} ⊆ CO.
Proof. For every M ∈ R-Mod by definitions we have:
(r ∨
α∈A
Cα
) (M) =
(
∨
α∈A
Cα
)
M
(0) =
∑
α∈A
[
(Cα)M(0)
]
=
=
∑
α∈A
[
rCα(M)] =
(
∨
α∈A
rCα
)
(M),
therefore r ∨
α∈A
Cα
=
∨
α∈A
rCα , proving the proposition.
90 Preradicals, closure operators in R-Mod
Thus the mapping Φ preserves the lattice operations “ ∧” and “∨ ”.
Now we consider the other two operations: product and coproduct.
Proposition 1.3. The mapping Φ converts the coproduct of CO into the
product of PR, i.e.
Φ (C # D) = Φ (C) · Φ (D)
for every operators C, D ∈ CO.
Proof. For every M ∈ R-Mod we have:
rC#D(M) = (C # D)M(0) = CDM (0)(0),
(rC · rD) (M) = rC
(
rD(M)
)
= rC
(
DM(0)
)
= CDM (0)(0).
Thus rC # D (M) = (rC · rD) (M) for every M ∈ R-Mod, i.e.
rC # D = rC · rD, as affirms the proposition.
Proposition 1.4. For every operators C, D ∈ CO the relation is true:
Φ (C · D) 6 Φ (C) : Φ (D).
If C is a maximal closure operator, then for every operator D ∈ CO
the equality holds:
Φ (C · D) = Φ (C) : Φ (D).
Proof. By definitions we have for every M ∈ R-Mod:
rC · D (M) = (C · D)M(0) = CM
(
DM(0)
)
,
[ (rC : rD) (M) ] / rD(M) = rC
(
M/ rD(M)
)
=
= rC
(
M/DM(0)
)
= CM/DM (0)(0̄).
Therefore [rC · D (M) ] / rD (M) =
[
CM
(
DM(0)
)]
/ DM(0). From the
definition of closure operator
(
condition (c3)
)
it follows that
[
CM
(
DM(0)
) ]
/ DM(0) ⊆ CM/DM (0)(0̄). So from the foregoing it fol-
lows that [rC · D (M)] / rD (M) ⊆ [(rC : rD) (M)] / rD(M). Hence
rC · D (M) ⊆ (rC : rD) (M) for every M ∈ R-Mod, i.e. rC · D 6 rC : rD,
which proves the first statement.
Now we suppose that the operator C is maximal, i.e.
[
CM(N)
]
/ N = CM/N(0̄) for every N ⊆ M . For N = DM(0)
we obtain
[
CM
(
DM(0)
)]
/ DM(0) = CM/DM (0)(0̄), which means that
[ rC · D (M) ]/ rD (M) = [ (rC : rD) (M) ] / rD (M). Therefore rC · D (M) =
(rC : rD) (M) for every M ∈ R-Mod, i.e. rC · D = rC : rD, proving the
second statement.
A. I. Kashu 91
2. The mapping Ψ1 and its impact on the operations of PR
In continuation we consider the mapping Ψ1 : PR → CO, which is
defined by the rule Ψ1(r) = C r and [ (C r)M(N) ] / N = r(M/ N) for
every r ∈ PR and N ⊆ M . We verify the influence of this mapping on
the operations of PR.
Proposition 2.1. The mapping Ψ1 preserves the operation of meet, i.e.
Ψ1
(
∧
α∈A
rα
)
=
∧
α∈A
[
Ψ1 (rα)
]
for every family of preradicals {rα|α ∈ A} ⊆ PR.
Proof. By definitions for every N ⊆ M we have:
[(
C
∧
α∈A
rα
)
M
(N)
]
/ N =
(
∧
α∈A
rα
)
(M/ N) =
⋂
α∈A
[
rα(M/ N)
]
=
=
⋂
α∈A
[(
(C rα)M(N)
)
/ N
]
=
[
⋂
α∈A
(
(C rα)M(N)
)]
/ N =
[(
∧
α∈A
C rα
)
M
(N)
]
/ N .
Hence
(
C
∧
α∈A
rα
)
M
(N) =
(
∧
α∈A
C rα
)
M
(N) for every N ⊆ M , i.e.
C
∧
α∈A
rα
=
∧
α∈A
C rα , which proves the proposition.
Proposition 2.2. The mapping Ψ1 preserves the operation of join, i.e.
Ψ1
(
∨
α∈A
rα
)
=
∨
α∈A
[
Ψ1 (rα)
]
for every family of preradicals {rα|α ∈ A} ⊆ PR.
Proof. For every N ⊆ M by definitions we have:
[(
C
∨
α∈A
rα
)
M
(N)
]
/ N =
(
∨
α∈A
rα
)
(M/ N) =
∑
α∈A
[
rα(M/ N)
]
=
=
∑
α∈A
[(
(Crα)M(N)
)
/ N
]
=
[
∑
α∈A
(
(C rα)M(N)
)]
/ N =
[(
∨
α∈A
C rα
)
M
(N)
]
/ N .
Hence
(
C
∨
α∈A
rα
)
M
(N) =
(
∨
α∈A
C rα
)
M
(N) for every N ⊆ M , which
means that C
∨
α∈A
rα
=
∨
α∈A
C rα , proving the proposition.
92 Preradicals, closure operators in R-Mod
Remark. Since the mappings Φ and Ψ1 preserve the lattice operations
“∧” and “∨ ”, the bijection M ax (CO) ∼= PR, which is defined by these
mappings, is an isomorphism of complete “big” lattices.
Further we consider the effect of the mapping Ψ1 on the rest of
operations: product and coproduct.
Proposition 2.3. The mapping Ψ1 converts the product of PR into the
coproduct of CO, i.e.
Ψ1 (r · s) = Ψ1 (r) # Ψ1 (s)
for every preradicals r, s ∈ PR.
Proof. Let r, s ∈ PR and N ⊆ M . Then:
[(
C r · s
)
M
(N)
]
/ N = (r · s) (M/ N) = r
(
s(M/ N)
)
.
From the other hand, by the definition of coproduct in CO we obtain:
[
(C r # C s)M(N)
]
/ N =
[
(C r)(C s)M (N)(N)
]
/ N =
= r
[(
(C s)M(N)
)
/ N
]
= r
(
s(M/ N)
)
.
Therefore
[(
C r · s
)
M
(N)
]
/ N =
[
(C r # C s)M(N)
]
/ N , and so
(C r · s)M(N) = (C r # C s)M(N) for every N ⊆ M . This means that
C r · s = C r # C s, proving the proposition.
Proposition 2.4. The mapping Ψ1 converts the coproduct of PR into
the product of CO, i.e.
Ψ1 (r : s) = Ψ1 (r) · Ψ1 (s)
for every preradicals r, s ∈ PR.
Proof. By definition [ (C r:s)M(N) ] / N = (r : s) (M/ N) and
[
(r : s) (M/ N)
]
/ s(M/ N) = r
[
(M/ N) / s(M/ N)
]
=
=
[
(C r)M/N
(
s(M/ N)
)]
/ s(M/N) for every N ⊆ M .
Therefore (r : s) (M/ N) = (C r)M/N
(
s(M/ N)
)
, i.e.
[(
C r:s
)
M
(N)
]
/ N =
(C r)M/N
(
s(M/ N)
)
.
Now we will use the fact that C r is a maximal closure operator (see
Section 1), which in the situation N ⊆ (C s)M(N) ⊆ M implies the
relations:
[
(C r)M
(
(C s)M(N)
)]
/ N = (C r)M/N
[(
(C s)M(N)
)
/ N
]
=
= (C r)M/N
(
s(M/ N)
)
.
A. I. Kashu 93
Comparing with the relation obtained above, now we have:
[(
C r:s
)
M
(N)
]
/ N =
[
(C r)M
(
(C s)M(N)
)]
/ N,
thus
(
C r:s
)
M
(N) = (C r · C s)M(N) for every N ⊆ M . This means that
C r:s = C r · C s, proving the proposition.
3. The mapping Ψ2 : impact on the operations of PR
Finally, we study the mapping Ψ2 : PR → CO, where Ψ2(r) = Cr and
(Cr)M(N) = N + r(M) for every r ∈ PR and N ⊆ M . We show the
influence of Ψ2 to the operations of PR.
Proposition 3.1. For every family of preradicals {rα|α ∈ A} ⊆ PR the
relation is true:
Ψ2
(
∧
α∈A
rα
)
6
∧
α∈A
[
Ψ2 (rα)
]
.
If the lattice L(M) is infinite distributive (relative to meets) for every
M ∈ R-Mod, then the equality holds:
Ψ2
(
∧
α∈A
rα
)
=
∧
α∈A
[
Ψ2 (rα)
]
.
Proof. By definitions Ψ2
(
∧
α∈A
rα
)
= C∧
α∈A
rα
, where
(
C∧
α∈A
rα
)
M
(N) =
N + [ (
∧
α∈A
rα) (M) ] = N + [
⋂
α∈A
rα (M) ] for every N ⊆ M .
From the other hand,
∧
α∈A
[
Ψ2 (rα)
]
=
∧
α∈A
Crα , where
(
∧
α∈A
Crα
)
M
(N) =
⋂
α∈A
[
(Crα)M(N)
]
=
⋂
α∈A
[
N + rα(M)
]
for every N ⊆ M . Since N +
[
⋂
α∈A
rα(M)
]
⊆
⋂
α∈A
[
N + rα(M)
]
, we have
(
C∧
α∈A
rα
)
M
(N) ⊆
(
∧
α∈A
Crα
)
M
(N), i.e. C ∧
α∈A
rα
6
∧
α∈A
Crα , proving the first
statement.
If L(M) is infinite distributive (relative to meets) for every
M ∈ R-Mod, then N +
[
⋂
α∈A
rα(M)
]
=
⋂
α∈A
[N + rα(M)] for every
N ⊆ M , which implies the equality C∧
α∈A
rα
=
∧
α∈A
Crα , proving the second
statement.
94 Preradicals, closure operators in R-Mod
Proposition 3.2. The mapping Ψ2 preserves the operation of join, i.e.
Ψ2
(
∨
α∈A
rα
)
=
∨
α∈A
[
Ψ2 (rα)
]
for every family of preradicals {rα|α ∈ A} ⊆ PR.
Proof. For every N ⊆ M we have:
(
C ∨
α∈A
rα
)
M
(N) = N +
[(
∨
α∈A
rα
)
(M)
]
= N +
[
∑
α∈A
rα(M)
]
.
From the other hand:
(
∨
α∈A
Crα
)
M
(N) =
∑
α∈A
[
(Crα)M(N)
]
=
∑
α∈A
[
N + rα(M)
]
=
= N +
[
∑
α∈A
rα(M)
]
.
Therefore
(
C ∨
α∈A
rα
)
M
(N) =
(
∨
α∈A
Crα
)
M
(N) for every N ⊆ M , i.e.
C ∨
α∈A
rα
=
∨
α∈A
Crα , which proves the proposition.
It remains to verify the impact of the mapping Ψ2 on the operations
of product and coproduct of PR.
Proposition 3.3. For every preradicals r, s ∈ PR the relation is true:
Ψ2 (r · s) 6 Ψ2 (r) # Ψ2 (s).
Proof. Let r, s ∈ PR and N ⊆ M . Then:
(
Cr · s
)
M
(N) = N + [ (r · s) (M) ] = N + r
(
s(M)
)
,
(Cr # Cs)M(N) = (C r)(Cs)M (N)(N) = (Cr)N+s(M)(N) =
= N + r
(
N + s(M)
)
.
Since N + r
(
s(M)
)
⊆ N + r
(
N + s(M)
)
, we have (Cr · s)M(N) ⊆
(Cr # Cs)M (N) for every N ⊆ M , i.e. Cr · s 6 Cr # Cs, proving the
proposition.
Proposition 3.4. For every preradicals r, s ∈ PR the relation is true:
Ψ2 (r : s) > Ψ2 (r) · Ψ2 (s).
If the preradical r ∈ PR is cohereditary, then for every s ∈ PR the
equality holds:
Ψ2 (r : s) = Ψ2 (r) · Ψ2 (s).
A. I. Kashu 95
Proof. If r, s ∈ PR and N ⊆ M , then (Cr:s)M(N) = N + [(r : s) (M)] and
[(r : s) (M)] / s(M) = r
(
M / s(M)
)
. Therefore:
[ (Cr:s)M(N) ] / s(M) = [ N + (r : s) (M) ] / s(M) =
=
[(
N + s(M)
)
/ s(M)
]
+
[(
(r : s) (M)
)
/ s(M)
]
=
=
[(
N + s(M)
)
/ s(M)
]
+
[
r
(
M/ s(M)
)]
.
On the other hand, for the product of respective operators we have:
(Cr · Cs)M(N) = (Cr)M [ (Cs)M(N) ] =
= (Cr)M [N + s(M) ] = N + s(M) + r(M).
Hence:
[ (Cr · Cs)M(N) ] / s(M) = [N + s(M) + r(M)] / s(M) =
=
[(
N + s(M)
)
/ s(M)
]
+
[(
r(M) + s(M)
)
/ s(M)
]
.
From the definition of preradical we have the inclusion:
[r(M) + s(M)] / s(M) ⊆ r
(
M / s(M)
)
,
and comparing with the previous relations we obtain:
[ (Cr:s)M(N) ] / s(M) ⊇ [ (Cr · Cs)M(N) ] / s(M).
Therefore (Cr:s)M(N) ⊇ (Cr · Cs)M(N) for every N ⊆ M , which means
that Cr:s > Cr · Cs, proving the first statement.
If we suppose that the preradical r ∈ PR is cohereditary, then
for every s ∈ PR by previous calculation we obtain the equality
[(Cr:s)M(N) ] / s(M) = [ (Cr · Cs)M(N) ] / s(M), which implies the equal-
ity Cr:s = Cr · Cs.
References
[1] D. Dikranjan, E. Giuli, Factorizations, injectivity and compactness in categories
of modules, Commun. in Algebra, v. 19, №1, 1991, pp. 45–83.
[2] D. Dikranjan, W. Tholen, Categorical structure of closure operators, Kluwer Aca-
demic Publishers, 1995.
[3] L. Bican, T. Kepka, P. Nemec, Rings, modules and preradicals, Marcel Dekker,
New York, 1982.
[4] A.I. Kashu, Radicals and torsions in modules, Kishinev, Ştiinţa, 1983 (in Russian).
[5] A.I. Kashu, Closure operators in the categories of modules. Part I, Algebra and
Discrete Mathematics, v. 15 (2013), №2, pp. 213–228.
[6] A.I. Kashu, Closure operators in the categories of modules. Part II, Algebra and
Discrete Mathematics, v. 16 (2013), №1, pp. 81–95.
96 Preradicals, closure operators in R-Mod
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: kashuai@math.md
Received by the editors: 09.07.2014
and in final form 09.07.2014.
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