Closure operators in the categories of modules. Part II (Hereditary and cohereditary operators)
This work is a continuation of the paper [1] (Part I), in which the weakly hereditary and idempotent closure operators of the category R-Mod are described. Using the results of [1], in this part the other classes of closure operators C are characterized by the associated functions F₁с and F₂с which...
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irk-123456789-1523102019-06-11T01:25:07Z Closure operators in the categories of modules. Part II (Hereditary and cohereditary operators) Kashu, A.I. This work is a continuation of the paper [1] (Part I), in which the weakly hereditary and idempotent closure operators of the category R-Mod are described. Using the results of [1], in this part the other classes of closure operators C are characterized by the associated functions F₁с and F₂с which separate in every module M ∈ R-Mod the sets of C-dense submodules and C-closed submodules. This method is applied to the classes of hereditary, maximal, minimal and cohereditary closure operators. 2013 Article Closure operators in the categories of modules. Part II (Hereditary and cohereditary operators) / A.I. Kashu // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 81–95. — Бібліогр.: 9 назв. — англ. 1726-3255 2010 MSC:16D90, 16S90, 06B23. http://dspace.nbuv.gov.ua/handle/123456789/152310 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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This work is a continuation of the paper [1] (Part I), in which the weakly hereditary and idempotent closure operators of the category R-Mod are described. Using the results of [1], in this part the other classes of closure operators C are characterized by the associated functions F₁с and F₂с which separate in every module M ∈ R-Mod the sets of C-dense submodules and C-closed submodules. This method is applied to the classes of hereditary, maximal, minimal and cohereditary closure operators. |
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Kashu, A.I. |
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Kashu, A.I. Closure operators in the categories of modules. Part II (Hereditary and cohereditary operators) Algebra and Discrete Mathematics |
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Kashu, A.I. |
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Kashu, A.I. |
| title |
Closure operators in the categories of modules. Part II (Hereditary and cohereditary operators) |
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Closure operators in the categories of modules. Part II (Hereditary and cohereditary operators) |
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Closure operators in the categories of modules. Part II (Hereditary and cohereditary operators) |
| title_fullStr |
Closure operators in the categories of modules. Part II (Hereditary and cohereditary operators) |
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Closure operators in the categories of modules. Part II (Hereditary and cohereditary operators) |
| title_sort |
closure operators in the categories of modules. part ii (hereditary and cohereditary operators) |
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Інститут прикладної математики і механіки НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/152310 |
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Closure operators in the categories of modules. Part II (Hereditary and cohereditary operators) / A.I. Kashu // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 81–95. — Бібліогр.: 9 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
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AT kashuai closureoperatorsinthecategoriesofmodulespartiihereditaryandcohereditaryoperators |
| first_indexed |
2025-07-13T02:48:09Z |
| last_indexed |
2025-07-13T02:48:09Z |
| _version_ |
1837498257060659200 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 16 (2013). Number 1. pp. 81 – 95
© Journal “Algebra and Discrete Mathematics”
Closure operators in the categories of modules
Part II (Hereditary and cohereditary operators)
A. I. Kashu
Abstract. This work is a continuation of the paper [1]
(Part I), in which the weakly hereditary and idempotent closure
operators of the category R-Mod are described. Using the results
of [1], in this part the other classes of closure operators C are
characterized by the associated functions F
C
1
and F
C
2
which
separate in every module M ∈ R-Mod the sets of C-dense sub-
modules and C-closed submodules. This method is applied to the
classes of hereditary, maximal, minimal and cohereditary closure
operators.
1. Introduction. Preliminary definitions and results
The present work is devoted to the study of the closure operators of the
module categories and it is a continuation of the paper [1] (Part I). On the
basis of the results of Part I, where the weakly hereditary and idempotent
closure operators of R-Mod are described, the characterizations of new
types of closure operators C are shown by means of associated functions
F
C
1
and F
C
2
which are defined by the sets of C-dense or C-closed sub-
modules. With this purpose the following types of closure operators of
R-Mod are studied: hereditary, weakly hereditary maximal, hereditary
maximal, minimal, cohereditary (i.e. the most important subclasses of
the classes of weakly hereditary or idempotent closure operators).
For every studied class of closure operators C the conditions to the
associated functions F
C
1
and F
C
2
are indicated which are necessary and
2010 MSC: 16D90, 16S90, 06B23.
Key words and phrases: ring, module, preradical, closure operator, dense sub-
module, closed submodule, hereditary (cohereditary) closure operator.
82 Closure operators in the categories of modules, I I
sufficient for C to belong to the investigated class. In the theory of radicals
the corresponding results consist in the characterization of diverse kinds
of preradicals by means of associated classes of torsion or torsion free
modules ([5, 7, 8, 9]).
In what follows we use some notions and results of Part I ([1]). As it
was mentioned in [1], our studies are based to the facts on closure operators
of R-Mod [2, 3, 4, 5, 6] and on the radicals in modules ([5, 7, 8, 9]).
Let R be a ring with unity and R-Mod be the category of unitary
left R-modules. We denote by L(RM) the lattice of submodules of the
module M ∈ R-Mod.
Definition 1.1. A closure operator of R-Mod is a function C which
associates to every pair N ⊆ M , N ∈ L(RM), a submodule of M , denoted
by CM(N) such that the following conditions are satisfied:
(c1) N ⊆ CM(N);
(c2) If N ⊆ P for N, P ∈ L(RM), then CM(N) ⊆ CM(P );
(c3) If f : M → M ′ is an R-morphism and N ⊆ M , then
f
(
CM(N)
)
⊆ CM′
(
f(N)
)
.
Denote by CO the class of all closure operators of R-Mod.
Definition 1.2. A closure operator C ∈ CO is called:
a) weakly hereditary, if CM(N) = CC
M
(N)(N) for every N ⊆ M ;
b) idempotent, if CM(N) = CM
(
CM(N)
)
for every N ⊆ M .
Definition 1.3. Let C ∈ CO. A submodule N ∈ L(R(M) is called:
a) C-dense in M , if CM(N) = M ;
b) C-closed in M , if CM(N) = N .
We use the following notations:
F
C
1
(M) is the set of all C-dense submodules of M ;
F
C
2
(M) is the set of all C-closed submodules of M .
Thus every closure operator C ∈ CO determines two functions F
C
1
and F
C
2
by which in some cases the operator C can be completely
characterized. In order to formulate some conditions on these functions,
we consider an abstract function F which for every module M ∈ R-Mod
separates in L(RM) a (non empty) set of submodules F(M), where F is
compatible with the isomorphisms and M ∈ F(M).
A. I. Kashu 83
Definition 1.4. An abstract function F of a category R-Mod is called
a function of type F1 if it satisfies the following conditions:
1) If N ∈ F(Mα), Mα ⊆ M (α ∈ A), then N ∈ F
(
∑
α∈A
Mα
)
;
2) If N ⊆ P ⊆ M and N ∈ F(P ), then for every K ⊆ M we have
N + K ∈ F(P + K);
3) If f : M → M ′ is an R-morphism and N ∈ F(M), then
f(N) ∈ F
(
f(M)
)
.
From the condition 2) it follows the property:
4) If N ⊆ P ⊆ M and N ∈ F(M), then P ∈ F(M).
For an abstract function F of R-Mod and for every N ⊆ M we denote:
(CF)M(N) =
∑
{
Mα ⊆ M
∣
∣ N ⊆ Mα, N ∈ F(Mα)
}
. (1.1)
The following statement describes the weakly hereditary closure ope-
rators of R-Mod by the abstract functions of type F1.
Theorem 1.1 ([1], Theorem 2.6). The mappings C 7−→ F
C
1
and
F 7−→ C F define a monotone bijection between the weakly hereditary
closure operators C of R-Mod and the abstract functions F of type F1.
In a similar way the characterization of idempotent closure operators
C of R-Mod is obtained, using the associated function F
C
2
.
Definition 1.5. An abstract function F of R-Mod is called a function
of type F2 if it satisfies the following conditions:
1∗) If Nα ∈ F(M), Nα ⊆ M (α ∈ A), then
⋂
α∈A
Nα ∈ F(M);
2∗) If N ⊆ P ⊆ M and N ∈ F(P ), then for every submodule
K ⊆ M we have N ∩ K ∈ F(P ∩ K);
3∗) If g : M → M ′ is an R-morphism and N ′ ∈ F
(
g(M)
)
, then
g−1(N ′) ∈ F(M).
Remark that from the condition 2∗) it follows the property:
4∗) If N ⊆ P ⊆ M and N ∈ F(M), then N ∈ F(P ).
84 Closure operators in the categories of modules, I I
For an abstract function F of R-Mod and N ⊆ M we denote:
(CF)M(N) = ∩ {Nα ⊆ M | N ⊆ Nα, Nα ∈ F(M)}. (1.2)
The following statement shows the description of the idempotent
closure operators of R-Mod by the abstract functions of type F2.
Theorem 1.2 ([1], Theorem 3.6). The mappings C 7−→ F
C
2
and
F 7−→ CF define an antimonotone bijection between the idempotent
closure operators of R-Mod and the abstract functions of type F2 of this
category. �
Using the previous results (Theorems 1.1 and 1.2) in a similar man-
ner the weakly hereditary idempotent closure operators of R-Mod are
described. For that the following property of transitivity is used:
5) = 5∗) If N ⊆ P ⊆ M, N ∈ F(P ) and P ∈ F(M), then
N ∈ F(M).
Theorem 1.3 ([1], Corollaries 4.3, 4.6). a) The mappings C 7−→ F
C
1
and
F 7−→ C F define a monotone bijection between the weakly hereditary
idempotent closure operators of R-Mod and the transitive functions of
type F1 of this category.
b) The mappings C 7−→ F
C
2
and F 7−→ CF define an antimonotone
bijection between the weakly hereditary idempotent closure operators
of R-Mod and the transitive functions of type F2 of this category. �
In continuation we will use the bijections of Theorems 1.1–1.3 with the
intention to characterize the other important classes of closure operators
by the abstract functions of types F1 or F2, associated to the studied
closure operators.
2. Hereditary closure operators
In this section we consider a subclass of the class of weakly hereditary
closure operators of R-Mod and give the description of such operators by
the abstract functions of type F1, using the Theorem 1.1.
Definition 2.1. A closure operator C ∈ CO is called hereditary if for
every submodules L ⊆ N ⊆ M the following relation holds:
CN(L) = CM(L) ∩ N. (2.1)
A. I. Kashu 85
If C ∈ CO is hereditary and N ⊆ M , then by (2.1) in the situation
N ⊆ CM(N) ⊆ M we have:
CCM (N)(N) = CM(N) ∩ CM(N) = CM(N),
thus C is weakly hereditary. So is true
Lemma 2.1. Every hereditary closure operator of R-Mod is weakly
hereditary. �
Therefore, every hereditary closure operator C ∈ CO can be completely
described by the associated function F
C
1
(Theorem 1.1), which is an
abstract function of type F1.
For an arbitrary weakly hereditary closure operator C ∈ CO we will
find necessary and sufficient condition to F
C
1
for the operator C to be
hereditary. With this purpose we consider the condition 4∗) mentioned
in the Section 1:
4∗) If N ⊆ P ⊆ M and N ∈ F(M), then N ∈ F(P ).
Proposition 2.2. Let C be an arbitrary weakly hereditary closure ope-
rator of R-Mod and F
C
1
be the associated function of type F1 (Theo-
rem 1.1). Then the operator C is hereditary if and only if the function
F
C
1
satisfies the condition 4∗).
Proof. (⇒) Let C be a hereditary closure operator of R-Mod,
N ⊆ P ⊆ M and N ∈ F
C
1
(M). Then CM(N) = M and from (2.1) it
follows that CP (N) = CM(N) ∩ P = M ∩ P = P , i.e. N ∈ F
C
1
(P ) and
F
C
1
satisfies the condition 4∗).
(⇐) Let C be a weakly hereditary closure operator of R-Mod and
the associated function F
C
1
satisfies the condition 4∗). Then C can be
re-established by F
C
1
(
see (1.1)
)
and for submodules L ⊆ N ⊆ M we have:
CM(L) =
∑
{Kα ⊆ M | L ⊆ Kα, L ∈ F
C
1
(Kα)},
CN(L) =
∑
{K ′
α
⊆ N | L ⊆ K ′
α
, L ∈ F
C
1
(K ′
α
)}.
Since F
C
1
satisfies the condition 1) (Definition 1.4), from the relations
L ∈ F
C
1
(Kα) (α ∈ A) it follows that L ∈ F
C
1
(
∑
α∈A
Kα
)
, where
∑
α∈A
Kα =
CM(L). Using the condition 4∗) in the situation L ⊆ CM(L) ∩ N ⊆
CM(L), from the relation L ∈ F
C
1
(
∑
α∈A
Kα
)
we conclude that
L ∈ F
C
1
(
CM(L) ∩ N
)
. Therefore the submodule CM(L) ∩ N , which is
contained in N and contains L, is one of the submodules K ′
α
from the defi-
nition of CN(L). This implies CM(L) ∩ N ⊆ CM(L), the inverse inclusion
being trivial. Thus the relation (2.1) holds, i.e. C is hereditary.
86 Closure operators in the categories of modules, I I
From Theorem 1.1 and Proposition 2.2 it follows
Corollary 2.3. The mappings C 7−→ F
C
1
and F 7−→ C F define a mono-
tone bijection between the hereditary closure operators of R-Mod and
the abstract functions of type F1, which satisfy the condition 4∗). �
3. Weakly hereditary maximal closure operators
In this section we consider the weakly hereditary maximal ([2]) closure
operators C of R-Mod and show the condition, satisfied by the respective
abstract functions F
C
1
.
Definition 3.1. A closure operator C ∈ CO is called maximal if for
every N ⊆ M the following relation holds:
CM(N) / N = CM/N(0̄) (3.1)
The role of such closure operators will be specified in other part of this
work, studying the relation between the closure operators and preradicals
of R-Mod. Now we remark that the maximal closure operators can be
described by the following condition: if K ⊆ N ⊆ M , then
CM(N) / K = CM/K(N/ K). (3.2)
Lemma 3.1. A closure operator C ∈ CO is maximal if and only if C
satisfies the condition (3.2).
Proof. (⇒) Let C be a maximal closure operator and K ⊆ N ⊆ M .
From (3.1), substituting M by M/K and N by N/K, we obtain
[CM/K(N/ K)]
/
(N/ K) = C(M/K) / (N/K)(
¯̄0). (3.3)
On the other hand, from the isomorphism (M/ K)
/
(N/ K) ∼= M/ N we
have [CM(N) / K]
/
(N/ K) ∼= CM(N) / N . From (3.1) and the mentioned
isomorphism it follows that
[CM(N) / K]
/
(N/ K) = C(M/K) / (N/K)(
¯̄0). (3.4)
From (3.3) and (3.4) now we have
[CM(N) / K]
/
(N/ K) = [CM/K(N/ K)]
/
(N/ K),
therefore CM(N) / K = CM/K(N/ K), i.e. (3.2) holds.
(⇐) If K = N , then (3.2) implies (3.1).
A. I. Kashu 87
In what follows we consider the weakly hereditary maximal closure
operators C and using the Theorem 1.1 we give the characterization
of such operators by the associated functions F
C
1
. For that we use the
following modification of the condition 3∗) (Definition 1.5):
3̄) If K ⊆ N ⊆ M and N/ K ∈ F(M/ K), then N ∈ F(M).
Proposition 3.2. Let C ∈ CO be a weakly hereditary closure operator.
Then C is maximal if and only if the associated function F
C
1
satisfies
the condition 3̄).
Proof. (⇒) Let C be a maximal closure operator and K ⊆N ⊆M . Then
(3.2) holds (Lemma 3.1). If N/ K ∈ F
C
1
(M/ K), then CM/K(N/ K) =
M/ K and by (3.2) we have CM(N) / K = M/ K, thus CM(N) = M , i.e.
N ∈ F
C
1
(M) and F
C
1
satisfies the condition 3̄).
(⇐) Let C be a weakly hereditary closure operator for which the
function F
C
1
satisfies 3̄). Then C can be re-established by F
C
1
and in the
situation K ⊆ N ⊆ M we have:
CM(N) =
∑
{Mα ⊆ M | N ⊆ Mα, N ∈ F
C
1
(Mα)},
CM/K(N/ K) =
∑
{M ′
α
/ K ⊆ M/ K
∣
∣ N/ K ⊆ M ′
α
/ K,
N/ K ∈ F
C
1
(M ′
α
/ K)}.
To prove the maximality of C it is sufficient to verify in (3.2) the inclusion:
CM(N) / K ⊇ CM/K(N/ K). (3.5)
From the relations N/ K ∈ F
C
1
(M ′
α
/ K) (α ∈ A) by the condition 3̄)
it follows that N ∈ F
C
1
(M ′
α
) (α ∈ A). Using the condition 1) of F
C
1
(Definition 1.4), now we have N ∈ F
C
1
(
∑
α∈A
M ′
α
)
. Thus
∑
α∈A
M ′
α
is one of the
submodules Mα from the definition of CM(N), therefore CM(N) ⊇
∑
α∈A
M ′
α
and CM(N) / K ⊇
(
∑
α∈A
M ′
α
)
/ K. But
(
∑
α∈A
M ′
α
)
/ K =
∑
α∈A
(M ′
α
/ K) =
CM/K(N/ K), so CM(N)/ K ⊇CM/K(N/ K), proving (3.5). Therefore C is
maximal.
Combining Theorem 1.1 and Proposition 3.2 we obtain
Corollary 3.3. The mappings C 7−→ F
C
1
and F 7−→ C F define a mono-
tone bijection between the weakly hereditary maximal closure opera-
tors of R-Mod and the abstract functions of type F1 which satisfies the
conditions 3̄). �
88 Closure operators in the categories of modules, I I
4. Hereditary maximal closure operators
Restricting the bijection of Corollary 3.3 and using the characterization
of hereditary closure operators (Corollary 2.3), we now obtain the following
result on the hereditary maximal closure operators of R-Mod.
Proposition 4.1. The mappings C 7−→ F
C
1
and F 7−→ C F define a
monotone bijection between the hereditary maximal closure operators
of R-Mod and the functions of type F1 of R-Mod, which satisfy the
conditions 4∗) and 3̄). �
It is interesting the fact that the hereditary maximal closure operators
of R-Mod can be described by well known sets of left ideals of R, namely
by the preradical filters (or left linear topologies) of R ([5, 7, 8, 9]).
Definition 4.1. A set E ⊆ L(RR) of left ideals of the ring R is called
preradical filter of R if it satisfies the following conditions:
(a1) If I ∈E and a∈R, then (I :a) ∈ E (where (I :a) = {r∈R | r a ∈ I});
(a2) If I ∈ E and I ⊆ J (J ∈ L(RR)), then J ∈ E;
(a3) If I, J ∈ E, then I ∩ J ∈ E.
Proposition 4.2. Let C be a hereditary maximal closure operator of
R-Mod. Then the set of left ideals
E
C = F
C
1
(RR) = {I ∈ L(RR) | CR(I) = R}
is a preradical filter of R.
Proof. (a1) Let I ∈ F
C
1
(RR) and a ∈ R. Consider the R-morphism:
f : RR −→ (R a + I) / I ⊆ R / I, f(r) = ra + I ∀ r ∈ R.
Since Ker f = {r ∈ R | ra ∈ I} = (I : a), we have
R / (I : a) ∼= (R a + I) / I ⊆ R / I.
Using the condition 4∗) for F
C
1
in the situation I ⊆ R a + I ⊆ R, from
the relation I ∈ F
C
1
(RR) we obtain I ∈ F
C
1
(R a+I) and by the mentioned
isomorphism we conclude that (I : a) ∈ F
C
1
(RR).
(a2) Let I ∈ F
C
1
(RR) and I ⊆ J . Since F
C
1
satisfies the condition 2)
(Definition 1.4), it satisfies also the condition 4) which in the situation
I ⊆ J ⊆ R shows that the relation I ∈ F
C
1
(RR) implies J ∈ F
C
1
(RR).
(a3) Let I, J ∈ F
C
1
(RR). Consider the module M = (R / I) ⊕ (R / J)
and the R-morphism:
A. I. Kashu 89
f : RR −→ RM, f(r) = (r + I, r + J) ∀ r ∈ R.
Then Ker f = I ∩ J and R / (I ∩ J) ∼= Im f ⊆ M . From the assump-
tion I, J ∈ F
C
1
(RR) it follows that 0̄ ∈ F
C
1
(R / I) and 0̄ ∈ F
C
1
(R / J)
(
by
condition 3)
)
, therefore the condition 1) implies 0̄ ∈ F
C
1
(M). Now we
apply the condition 4∗) in the situation 0̄ ⊆ Im f ⊆ M and from the
relation 0̄ ∈ F
C
1
(M) we conclude that 0̄ ∈ F
C
1
(Im f). From the indicated
isomorphism it follows that 0̄ ∈ F
C
1
(
R / (I ∩ J)
)
. Since C is maximal,
F
C
1
satisfies the condition 3̄) (Proposition 3.2) which shows now that
I ∩ J ∈ F
C
1
(RR).
Proposition 4.3. Let E ⊆ L(RR) be an arbitrary preradical filter
of R and
(C E)M(N) = {m ∈ M | (N : m) ∈ E} (4.1)
where N ⊆ M and (N : m) = {r ∈ R | r m ∈ N}. Then C E is a
hereditary maximal closure operator of R-Mod.
Proof. From the conditions (a1)–(a3) of the Definition 4.1 it is obvious
that the rule (4.1) defines a submodule of M , containing N . The monotony
of C E also follows from the definitions.
To verify the condition (c3) (Definition 1.1) let f : M → M ′ be an
R-morphism and N ⊆ M . If m ∈ (C E)M(N), then (N : m) ∈ E and
(N : m) ⊆ (f(N) : f(m)), therefore from (a2) we have (f(N) : f(m)) ∈E.
Thus f(m) ∈ (C E)
f(M)
(f(N)) ⊆ C E
M′ (f(N)) and C E is a closure
operator of R-Mod.
Moreover, C E is hereditary: if L ⊆ N ⊆ M , then CN(L) = CM(L) ∩ N ,
since n ∈ N and n ∈ CM(L) imply (L : n) ∈ E and n ∈ CN(L).
Finally, we verify the maximality of C E. Let N ⊆ M . From (4.1) we
have:
(C E)
M/N
( 0̄ ) = {m + N ∈ M/ N |
(
0̄ : (m + N)
)
= (N : m) ∈ E},
[(C E)M(N)] / N = {m + N ∈ M/ N | m ∈ (C E)M(N)} =
= {m + N ∈ M/ N | (N : m) ∈ E}.
Therefore [(C E)M(N)] / N = (C E)
M/N
( 0̄ ), so C E is maximal
by (3.1).
Proposition 4.4.
a) If C is a hereditary maximal closure operator of R-Mod, then
C = CE
C
.
b) If E is a preradical filter of R, then E = E
CE
.
90 Closure operators in the categories of modules, I I
Proof. a) To verify the inclusion CM(N) ⊆
(
CE
C )
M
(N) let N ⊆ M
and m ∈ CM(N). From the isomorphism:
R/(N : m)
ϕ
∼= (R m+N)/ N ⊆ M/ N, ϕ (r + (N : m)) = r m+N ∀ r ∈ R
we have (N : m) ∈ F
C
1
(RR) if and only if N ∈ F
C
1
(R m + N). Since
by assumption C is hereditary, it is weakly hereditary
(
CC
M
(N)(N) =
CM(N)
)
, therefore N ∈ F
C
1
(CM(N)). By the hereditary of C we have
also the condition 4∗) for F
C
1
(Proposition 2.2) which in the situation
N ⊆ R m + N ⊆ CM(N), N ∈ F
C
1
(
CM(N)
)
implies N ∈ F
C
1
(R m + N).
By the previous remark this means that (N : m) ∈ F
C
1
(RR), i.e.
m ∈
(
CE
C )
M
(N), proving that CM(N) ⊆
(
CE
C )
M
(N).
For the inverse inclusion let m ∈ (CE
C
)M(N), i.e. (N : m) ∈ F
C
1
(RR).
From the mentioned isomorphism we have N ∈ F
C
1
(R m + N). Since C is
weakly hereditary, it can be expressed by the function F
C
1
as follows:
CM(N) =
∑
{Mα ⊆ M | N ⊆ Mα, N ∈ F
C
1
(Mα)}.
From the relation N ∈ F
C
1
(R m + N) it is clear that R m + N is one
of Mα from the definition of CM(N). Therefore R m + N ⊆ CM(N), i.e.
m ∈ CM(N), proving the needed inclusion. This means that C = CE
C
.
b) By definitions we have:
E
CE
= F
CE
1 (RR) = {I ∈ L(RR) | (CE)R(I) = RR} =
= {I ∈ L(RR) | (I : r) ∈ E ∀ r ∈ R}.
If I ∈ E, then by the condition (a1) we have (I : r) ∈ E for every
r ∈ R, i.e. I ∈ E
CE
, proving that E ⊆ E
CE
.
If I ∈ E
CE
, then (I : r) ∈ E for every r ∈ R, so (I : 1R) = I ∈ E. Thus
E
CE
⊆ E.
From the Propositions 4.2–4.4 we obtain
Corollary 4.5. The mappings C 7−→ E
C and E 7−→ CE define a mono-
tone bijection between the hereditary maximal closure operators of
R-Mod and the preradical filters of the ring R. �
It is a well known fact that every preradical filter defines an unique
pretorsion (or: hereditary preradical) of R-Mod ([5, 7, 9]). Thus by Corol-
lary 4.5 there exists a monotone bijection between the hereditary maximal
closure operators of R-Mod and the pretorsions of this category. Other
method of proving this result will be mentioned studying the relations of
CO with the preradicals of R-Mod.
A. I. Kashu 91
5. Minimal closure operators
In the previous studies the subclasses of the class of weakly hereditary
closure operators were considered: hereditary, weakly hereditary maximal,
hereditary maximal. Using the monotone bijection of Theorem 1.1, we
obtained the characterizations of these kinds of closure operators C by
means of the associated functions FC
1
. In continuation we will operate in a
similar manner, investigating some subclasses of the class of idempotent
closure operators of R-Mod. Using the Theorem 1.2 we will show the
characterizations of such types of closure operators C by means of the
functions F
C
2
, adding some new conditions to the set of conditions 1∗),
2∗), 3∗) (Definition 1.5).
Definition 5.1. A closure operator C ∈ CO is called minimal if
CM(N) = CM(0) + N (5.1)
for every N ⊆ M ([2]).
We indicate the other form of minimality of C, using the following
condition: if L ⊆ N ⊆ M , then
CM(N) = CM(L) + N. (5.2)
Lemma 5.1. A closure operator C ∈ CO is minimal if and only if it
satisfies the condition (5.2).
Proof. (⇒) If C ∈ CO is minimal and L ⊆ N ⊆ M , then CM(N) =
CM(0) + N and CM(L) = CM(0) + L, therefore
CM(L) + N = (CM(0) + L) + N = CM(0) + N = CM(N),
i.e. (5.2) is true.
(⇐) From (5.2) for L = 0 we obtain (5.1).
Lemma 5.2. Every minimal closure operator of R-Mod is idempotent.
Proof. If C ∈ CO is minimal, then for every N ⊆ M we have:
CM (CM(N)) = CM (CM(0) + N) = CM(0) + (CM(0) + N) =
= CM(0) + N = CM(N).
Therefore every minimal closure operator C can be described by the
corresponding function F
C
2
(Theorem 1.2), which in this case satisfies
the conditions 1∗), 2∗), 3∗) (Definition 1.5).
92 Closure operators in the categories of modules, I I
Let C be an arbitrary idempotent closure operator of R-Mod. Then it
is completely determined by the associated function F
C
2
. It is natural the
question: what condition for F
C
2
must be added to 1∗), 2∗), 3∗) for the
operator C to be minimal? The answer can be obtained by the condition 4)
of Section 1:
4) If N ⊆ P ⊆ M and N ∈ F(M), then P ∈ F(M).
Proposition 5.3. Let C be an idempotent closure operator of R-Mod.
Then the operator C is minimal if and only if the function F
C
2
satisfies
the conditions 4).
Proof. (⇒) Let C be a minimal closure operator, N ⊆ P ⊆ M and
N ∈ F
C
2
(M). Then CM(N) = N, CM(0) + N = N and CM(0) ⊆ N .
Therefore CM(P ) = CM(0) + P ⊆ N + P = P , so CM(P ) = P , i.e.
P ∈ F
C
2
(M) and F
C
2
satisfies 4).
(⇐) Suppose that C is idempotent and F
C
2
satisfies condition 4).
Then C can be re-established by F
C
2
and for L ⊆ N ⊆ M we have:
CM(N) = ∩ {Nα ⊆ M | N ⊆ Nα, Nα ∈ F
C
2 (M)},
CM(L) = ∩ {Lα ⊆ M | L ⊆ Lα, Lα ∈ F
C
2
(M)}.
Since F
C
2
satisfies the condition 1∗) (Definition 1.5), from the relations
Lα ∈ F
C
2
(M) (α ∈ A) it follows that
⋂
α∈A
Lα ∈ F
C
2
(M), i.e. CM(L) ∈
F
C
2
(M). Using the condition 4) in the situation CM(L) ⊆ CM(L) + N ⊆
M , from the relation CM(L) ∈ F
C
2
(M) we have CM(L) + N ∈ F
C
2
(M).
Therefore CM(L) + N is one of the submodules Nα from the definition
of CM(N), so CM(N) ⊆ CM(L) + N , the inverse inclusion being trivial.
Thus (5.2) holds, i.e. C is minimal.
From Theorem 1.2 and Proposition 5.3 it follows
Corollary 5.4. The mappings C 7−→ F
C
2
and F 7−→ CF define an
antimonotone bijection between the minimal closure operators of R-Mod
and the abstract functions of type F2, which satisfy the condition 4). �
In the previous situation if we add for C the condition to be weakly
hereditary, then for the function F
C
2
we must join the condition of transi-
tivity 5) = 5∗)
(
Theorem 1.3 b)
)
. In such way by the restriction of the
bijection of Corollary 5.4 we obtain
Corollary 5.5. The mappings C 7−→ F
C
2
and F 7−→ CF define an an-
timonotone bijection between the weakly hereditary minimal closure
operators of R-Mod and the transitive functions of type F2 which satisfies
the conditions 4). �
A. I. Kashu 93
6. Cohereditary closure operators
In this section we will consider a new class of closure operators C which
is a subclass of idempotent operators, therefore such operators possess
the characterization by the associated functions F
C
2
(Theorem 1.2).
Definition 6.1. A closure operator C ∈ CO will be called cohereditary
if for every R-morphism f : M → M ′ of R-Mod and every N ⊆ M the
following relation holds:
f
(
CM(N)
)
= Cf(M)
(
f(N)
)
. (6.1)
The other form of this condition is the following: for every submodules
K, N ∈ L(RM) is true the relation
(
CM(N) + K
)
/ K = CM/K
(
(N + K) / K
)
. (6.2)
Lemma 6.1. Every cohereditary closure operator C ∈ CO is minimal,
therefore it is also idempotent.
Proof. Let C ∈ CO be cohereditary and N ⊆ M . Applying (6.1) to the
natural morphism πN : M → M/ N and N ⊆ CM(N) ⊆ M , we have
CM(N) / N = CM/N(0̄). By cohereditary for πN and 0 ⊆ CM(0) ⊆ M we
obtain
(
CM(0) + N
)
/ N = CM/N(0̄). Therefore CM(N)/ N =
(
CM(0) +
N
)
/ N and CM(N) = CM(0) + N , i.e. C is minimal. By the Lemma 5.2
C is idempotent.
Lemma 6.2. A closure operator C is cohereditary if and only if it is
maximal and minimal.
Proof. (⇒) Let C ∈ CO be cohereditary and K ⊆ N ⊆ M . Then
from (6.2) CM(N)/ K = CM/K(N/ K), so (3.2) holds, i.e. C is maximal
(Lemma 3.1). By the Lemma 6.1 C is also minimal.
(⇐) Let C ∈ CO be maximal and minimal, and K, N ∈ L(RM). The
minimality of C
(
see (5.2)
)
in the situation N ⊆ N + K ⊆ M implies
CM(N + K) = CM(N) + (N + K) = CM(N) + K,
therefore
(
CM(N) + K
)
/ K = CM(N + K) / K.
On the other hand, since C is maximal by (3.2) in the situation
K ⊆ N + K ⊆ M we have:
(
CM(N + K)
)
/ K = CM/K
(
(N + K) / K
)
.
Comparing with the previous relation we obtain
(
CM(N) + K
)
/ K = CM/K
(
(N + K) / K
)
,
i.e. C is cohereditary.
94 Closure operators in the categories of modules, I I
Let C ∈ CO be an arbitrary cohereditary closure operator. Since it is
idempotent (Lemma 6.1), it can be described by the associated function
F
C
2
(Theorem 1.2) which in this case satisfies the conditions 1∗), 2∗), 3∗)
(Definition 1.5).
For an arbitrary idempotent closure operator C ∈ CO we consider the
associated function F
C
2
and find condition to F
C
2
for C to be cohereditary.
For that we use the condition 3) of Definition 1.4:
3) If f : M → M ′ is an R-morphism and N ∈ F(M), then
f(N) ∈ F
(
f(M)
)
.
Proposition 6.3. Let C be an idempotent closure operator of R-Mod
and F
C
2
be the associated function (of type F2). Then C is cohereditary if
and only if the function F
C
2
satisfies the condition 3).
Proof. (⇒) Let C ∈ CO be cohereditary, f : M → M ′ be an
R-morphism and N ∈ F
C
2
(M). Then CM(N) = N and from (6.1)
f
(
CM(N)
)
= Cf(M)
(
f(N)
)
, i.e. f(N) = Cf(M)
(
f(N)
)
. Thus f(N) ∈
F
C
2
(
f(M)
)
and F
C
2
satisfies the condition 3).
(⇐) Let C ∈ CO be idempotent and F
C
2
satisfies the condition 3).
Then C can be expressed by the function F
C
2
. So for f : M → M ′ and
N ⊆ M we have:
CM(N) = ∩ {Nα ⊆ M | N ⊆ Nα, Nα ∈ F
C
2
(M)},
Cf(M)
(
f(N)
)
= ∩ {N ′
α
⊆ f(M) | f(N) ⊆ N ′
α
, N ′
α
∈ F
C
2
(
f(M)
)
}.
Since F
C
2
satisfies the condition 1∗) (Definition 1.5), from the relations
Nα ∈ F
C
2
(M) (α ∈ A) it follows that
⋂
α∈A
Nα ∈ F
C
2
(M). Now from the
condition 3) of F
C
2
we obtain f
(
⋂
α∈A
Nα
)
∈ F
C
2
(
f(M)
)
. Therefore the sub-
module f
(
⋂
α∈A
Nα
)
⊆ f(M) which contains f(N) is one the submodules
N ′
α
from the definition of Cf(M)
(
f(N)
)
. So Cf(M)
(
f(N)
)
⊆ f
(
⋂
α∈A
Nα
)
=
f
(
CM(N)
)
and the inverse inclusion is true by (c3) (Definition 1.1). This
proves that f
(
CM(N)
)
= Cf(M)
(
f(N)
)
, i.e. C is cohereditary.
From Theorem 1.2 and Proposition 6.3 it follows
Corollary 6.4. The mappings C 7−→ F
C
2
and F 7−→ CF define an an-
timonotone bijection between the cohereditary closure operators C of
R-Mod and the abstract functions F of type F2 which satisfy the condi-
tion 3). �
A. I. Kashu 95
If we limit the previous bijection to the weakly hereditary operators
C, then for the corresponding function F
C
2
we must add the condition of
transitivity 5) = 5∗)
(
Theorem 1.3 b)
)
, so we obtain
Corollary 6.5. The mappings C 7−→ F
C
2
and F 7−→ CF define an anti-
monotone bijection between the weakly hereditary and cohereditary
closure operators of R-Mod and the transitive functions of type F2 which
satisfy the condition 3). �
References
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and idempotent operators), Algebra and Discrete Mathematics, v. 15, №2, 2013,
pp. 213–228.
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modules, Commun. in Algebra, v. 19, №1, 1991, pp. 45–83.
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1987, pp. 129–143.
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Categorical Topology, Prague, 1988 (World Scientific Publ., Singapore, 1989).
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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: 03.06.2013
and in final form 03.06.2013.
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