西 南 交 通 大 学 学 报
第 54 卷 第 6 期
2019
年 12 月
JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY
Vol. 54 No. 6
Dec. 2019
ISSN: 0258-2724 DOI：10.35741/issn.0258-2724.54.6.51
Research Article Mathematics
S
TRONGLY
∩
–R
EVERSIBLE
E
XTENDING
A
CTS OVER
M
ONOIDS
∩-可逆扩展单调
Saad A. Al-Saadi, Darya J. Abdul Kareem, Angham A. Dahash Department of Mathematics, College of Science, Mustansiriya University
P.O. Box: 14022, Palestine St., Baghdad, Iraq, [email protected], [email protected], [email protected]
Abstract
In this work, we present and investigate the class of acts that are strongly extending relative to -reversible subacts in the category of S-acts with zero element. Many characterizations and properties of
-reversible extending acts are described. Many results which are known for strongly uniform extending modules and strongly uniform continuous modules are generalized to S-acts with zero element.
Keywords: -reversible extending act, uniform S-act, injective act
摘要 在这项工作中，我们介绍并研究在零元素小号行为类别中相对于-可逆子行为强烈扩展的行 为类别。描述了可逆延伸作用的许多表征和性质。对于强均匀扩展模块和强连续模块已知的许多 结果被推广为零元素的小号行为。 关键词: -可逆延伸动作，均匀小号动作，内射动作
I. I
NTRODUCTION ANDP
RELIMINARIESLet be a monoid with identity . A unitary (right) -act is a nonempty set associated with
a multiplication and
for all and . Note
that -acts over a monoid may occur under different names such as -systems, -automata, -sets, -polygons and others. An element in
is called a zero of if for each . A
nonempty subset of an -act is called a
subact of if for all and .
Assume that are two right -acts. A
mapping is called a homomorphism
(denoted -homomorphism) if
for all and . An -homomorphism
is called a retraction if there is
such that . In this case
is called a retract of [13]. In addition, an -act is a retract of if and only if there is a subact
of and an epimorphism such that
and for every [13]. In this
paper, we assume that is monoid with zero, all acts with unique zero and any subact of an -act has the zero . According to [13], proposition
2.1.15, the category of all -acts with unique zero
and -homomorphisms preserving zero,
denoted by Act , has complete coproducts
of any nonempty families of -acts. More
accurately, with
if in for all . In
a similar manner, if is an -act such ,
where and are subacts of , then one can
write and is called a direct summand
of . So, (from [13], definition 1.5.7), in this
situation is called a decomposition of
. Otherwise, is called indecomposable.
Following [5], a subact is a direct summand of if and only if is a retract of . A subact is called large (or is an essential extension of )
in an –act if for any -homomorphism
for any -act , such that the restriction
to is a monomorphism, then is itself a
monomorphism [10]. Moreover, a nonzero subact of , is called a large intersection (shortly, -
large) if , for all nonzero subact of
[10]. A nonzero act is called reversible [1] (or uniform [10]) if every nonzero subact is large. In addition, a nonzero act is called -reversible [1] (or -uniform) if every nonzero subact is -large. From [10], proposition 4.7, every large subact is
-large, but the opposite is not true in general. For more details, basic results and any undefined concepts of acts over monoids, we refer to [13].
Let be an associative ring with identity and a module a unitary right -module. Projective modules and injective modules play a milestone role in the categories of modules and rings. The class of extending modules is one of the important and well-known generalizations of the
class of injective modules. A module is
extending (or has condition) if every
submodule of is essential in a direct summand of . This is equivalent to saying that every closed submodule of is direct summand (this condition is known as CS-module). For more
details of extending modules and their
generalizations, we refer to [9] and [15]. A submodule of a module is called essential if
the intersection of with any nonzero
submodule of is nonzero. A nonzero module is called uniform if every nonzero submodule of is essential. A submodule of a module is
called closed if has no proper essential
extension in . A module is called uniform-extending if every uniform submodule is
essential in a direct summand of [15].
Equivalently, a module is uniform-extending if, and only if, every uniform closed submodule is the direct summand of [15]. In other directions, as a strong concept of extending property, strongly extending modules have been studied in
[3], and later in [8]. A module is strongly extending if every submodule is essential in a fully-invariant direct summand of . Recently, in [4] a module is said to be strongly uniformly extending if every uniform submodule of is essential in a fully invariant direct summand of .
In a recent work [6], the author gave the version of extending modules in the category of acts over monoids. An -act is called extending if every subact of is -large in retract of . Following [7], an -act is called a strongly extending act if every subact is large in a fully invariant (stable) retract. A subact of an -act is called fully invariant (resp. stable) if
for each -homomorphism
[14] (resp. [11]). From [2], every stable subact of an act is fully invariant but not conversely in general. Also from [7], every retract fully invariant subact is stable. We symbolize as the disjoint union of sets and .
II. S
TRONGLY-R
EVERSIBLEE
XTENDING-A
CTSThe major aim of this work is investigation of the class of acts that are strongly extending relative to -reversible subacts in the category of
-acts with zero element.
A. Definition 2.1
An -act is called (strongly) -reversible extending if every intersection of reversible subacts of is -large in a (fully invariant) retract of . For easy terminology, we will write (strongly) -reversible extending acts instead of (strongly) intersection-reversible extending.
It is obvious that -reversible -acts and strongly extending acts are proper examples of strongly -reversible extending -acts. In other
directions, strongly -reversible extending
property is properly stronger than -reversible extending property.
B. Example 2.2
1. Consider as -act where is
monoid with multiplication. It is easily verifiable that is -reversible extending -act, while it is not strongly -reversible extending.
2. Let as -act where is the
monoid of all real numbers with multiplication. It is clear that there is no -reversible subacts of and so is strongly -reversible extending. But
is not strongly extending -act since is
closed subact of while it is not fully invariant. 3. Every -reversible -act is -reversible extending, but not conversely in general.
Consider as -act where is monoid with
multiplication. is a strongly -reversible
extending -act but is not -reversible.
The next lemma gives us useful properties for -reversible acts and retract subacts. The proof is straightforward by using the definitions.
C. Lemma 2.3
1. The -reversible acts are preserved by isomorphic property.
2. Every essential extension of a -reversible act is -reversible.
3. Let and be -acts such that
. Then is retract in if and only if is retract in and is retract in .
The following result is found in [11].
D. Lemma 2.4
Let and are -acts such that
. Then is -large in if and only if is -large in and is -large in .
The ideas of extending modules in [9] inspire us to get the following result for strongly -reversible extending acts.
E. Proposition 2.5
An -act is strongly -reversible
extending if and only if every -reversible closed subact of is a fully invariant retract of .
Proof: Let be strongly -reversible extending -act and let W be a -reversible closed subact . Then there exists a fully invariant retract of such that is -large in
. But is a closed subact of , hence
and so is a fully invariant retract of Conversely, let be a -reversible subact of . Then, by Zorn’s lemma, there exists a closed
subact of such is -large in . By
lemma 2.3 [2], then is -reversible closed in , so, by assumption, is a fully invariant retract of . Then, is strongly -reversible extending.
The following result gives an equivalent property of strongly reversible extending -acts.
F. Theorem 2.6
An -act is strongly -reversible
extending if and only if, for each -reversabile subact of , there is a disjoint union
such that , with a fully
invariant subact of and being -large
of .
Proof: Take as a strongly -reversible extending -act. Let be a -reversible of . Then is -large in fully invariant retract of
. Let , where is a subact of .
Moreover, since is -large in and it is
-large in ; thus, is -large in
. Conversely, let be a -reversible subact of . By hypothesis, there is a disjoint
union such that where
is a fully invariant subact of and is
-large in . We claim that is -large in . Let be a non-zero subact of , from now is a
subact of ; therefore, . This
implies that there exists a nonzero element
, and hence , then
, (since ). Hence and
then . Therefore, is -large in .
Thus, is a strongly reversible extending -act.
Following [7], each fully invariant retract of
an -act is stable. So, we have the next
characterization of strongly -reversible
extending acts.
G. Proposition 2.7
An -act is strongly -reversible extending if and only if every -reversible subact of is
-large in a stable retract of .
The next result asserts that the strongly -reversible extending property is closed under retracts.
H. Proposition 2.8
A closed subact of a strongly -reversible extending act is strongly -reversible extending.
Proof: Assume that is a closed subact of a
strongly -reversible extending -act . Take
to be a -reversible closed subact of . As is a closed subact of , so is a -reversible closed
subact of [6] (Lemma 2.4). Since is
strongly -reversible extending, then is a fully
invariant retract of . Currently, since ,
then is a retract of . Let be any
-endomorphism of . Then, one can form the
sequence , where is the
inclusion mapping and is the projection
homomorphism. So it is easy to see that is fully invariant of . Therefore, is strongly -reversible extending.
I. Corollary 2.9
A retract of a strongly -reversible extending -act is strongly -reversible extending.
J. Proposition 2.10
Each subact of a strongly -reversible
extending -act , where its intersection with any fully invariant retract of , is a fully
invariant retract in , which is strongly -reversible extending .
Proof: Assume that is an -reversible
subact of . As is strongly -reversible
extending, then there is a fully invariant retract
of such that is -large in . But
, thus by [6] is -large in
and, by assumption, is a fully
invariant retract in . Therefore, is strongly -reversible extending.
Now, we give details that a disjoint union of strongly -reversible extending acts need to not be strongly -reversible extending. In fact, is not a strongly -reversible extending
-act, while is a strongly -reversible
extending -act. The next theorem examines
when the class of strongly -reversible extending acts is close under disjoint union property.
K. Theorem 2.11
Let where and are -acts.
Then is strongly -reversible extending if and
only if i) and are strongly -reversible
extending -acts, and ii) every -reversible
closed subact of with or
is a fully invariant retract.
Proof: The necessary part is obtained directly
from corollary 2.9 and proposition 2.5. For the sufficient part, assume that Z is a -reversible closed subact of . By Zorn’s lemma, there
exists a complement subact of
where is -large in . As Z is a closed subact of , therefore is a closed subact of by [6],
lemma 2.4. Obviously, since is
-large in then . Hence, by
assumption, for some subact of
and is a fully invariant retract of . Now, . So
is closed in since is a closed
subact of . Also, . By
hypothesis, is a fully invariant retract of
and therefore of (since, .
Thus, where is a subact of .
Now .
Also, and are fully invariant retracts of
and , so is fully invariant
of . So is a fully invariant retract of .
Theretofore, is strongly -reversible
extending.
L. Proposition 2.12
Let be an -act, where is a
subact of for each . Then, the next
statements are equivalent:
1. is strongly -reversible extending; 2. All is strongly -reversible extending and every -reversible closed subact of is fully invariant;
3. Every is -reversible extending and all
-reversible closed subacts of are fully
invariant.
Proof: . By using corollary 2.9 together with proposition 2.5.
. It is clear.
. Assume that is a -reversible
closed subact of and is the
projection of -homomorphism on .
Take , so where . Then
is a -reversible closed subact of , and then, by assumption, is fully invariant and hence
. So and
hence . Thus .
Also, one can simply see that .
Therefore, . Now, is a
retract of and is closed in . Since is
-reversible, so is -reversible. But is
-reversible closed in , hence is
-reversible closed in [6]. Since
, then is -reversible
closed in . Using -reversible extending
property of , is a retract of . So
is a retract of , and so is
a fully invariant retract of . Hence, is strongly -reversible extending.
From [7], an -act is -reversible if and
only if is (strongly) extending and
indecomposable. Furthermore, we stated that each -reversible act is strongly -reversible extending, but not conversely, in general (Example 2.2 [3]). We have no idea whether there is a comparable result for strongly -reversible extending acts. In fact, the next result arises for -reversible extending acts.
M. Proposition 2.13
Let be an -act that contains a -reversible subact. Then, is -reversible if and only if is strongly -reversible and indecomposable.
Proof: Assume that is a -reversible subact of . By Zorn’s lemma, there exists a closed subact of such that is -large in . By lemma 2.3 [2], thus is a -reversible subact
of . Now, by the strongly -reversible
extending property of , so is a fully invariant
retract of . Since is indecomposable and
, therefore . Therefore, is
-reversible.
Note that the indecomposability property in the above proposition is necessary. For example,
as a -act is strongly -reversible extending,
which has -reversible subact , while is
not a -reversible -act.
N. Definition 2.14
An -act is known as a duo if every -reversible retract of is fully invariant.
Evidently, every duo -act is -duo, but the converse is not true. The as a -act is -duo, which is not duo.
Then the next result is given a
characterization of strongly -reversible
extending acts by using - duo acts.
O. Proposition 2.15
An -act is strongly -reversible
extending if and only if is -reversible
extending and a -duo -act.
Proof: It is evidently just using definitions.
III.
-R
EVERSIBLEC
ONTINUOUSA
CTSThe classes of continuous modules and quasi-continuous module were initiated by Jeremy [12] and Mohamed et al. [14], respectively. In the modules, is known as continuous if satisfies
the conditions: Each submodule of is
essential in a direct summand. And Each
submodule of , which is isomorphic to the direct summand of is a direct summand. Also, a module is known as quasi-continuous if
satisfies the conditions and : The sum
of any two direct summands of with zero
intersection is a direct summand. Moreover, from [4], a module is known as 1-continuous if
satisfies the conditions and ,
which are : Each uniform submodule of
is essential in a direct summand and :
Each uniform submodule isomorphic to the direct summand of , which is itself a direct summand of .
The above concepts motivate us to give the
following conditions for an -act : : If
each -reversible subact of is -large in
retract.
: Each -reversible subact of ,
which is isomorphic to a retract of , is a retract
of .
: If two -reversible retracts have zero intersection, then their union is a retract.
A. Definition 3.1
An -act is called -reversible continuous (or 1-continuous), if it satisfies the conditions
and .
B. Definition 3.2
An -act is called -reversible
quasi-continuous (or 1-quasi-quasi-continuous), if it satisfies
the conditions and .
C. Lemma 3.3
If an -act has -condition, then it
achieves the condition.
Proof: Assume that and are two
-reversible retracts of such that .
Let be a subact of such that .
Let be the projection homomorphism
of onto . Then is a
monomorphism and so . But is a
-reversible subact, so is -reversible (from
lemma (2.3) [2]), hence . But
and are retracts of , by the
property, so is a retract of . This
implies that for some subacts
of . Now, Hence, so, by [3], thus . But and , hence is a
retract of . This implies that is a retract
of , thus has condition.
From lemma 3.3, one can assert that every
-reversible continuous module is -reversible
quasi-continuous. The converse of lemma 3.3
does not hold in general. For instance, a -act
is -reversible quasi-continuous, but it is not
-reversible continuous since is a
-reversible subact of as a -act and , but
is not a retract of as a -act.
D. Proposition 3.4
A retract of a reversible contiuous act is -reversible continuous.
Proof: Assume that is a retract of a
-reversible continuous -act . Let be a subact
of . Since has condition, thus there
is a retract of such that is -large in .
Let where is a subact of . In
order, to show that has the condition ,
assume that is a -reversible subact of such that is isomorphic to the retract of . Set for some subact of . Furthermore,
take for some subact of . Then,
we have
.
condition so is a retract of . Since
, then is a retract of (by lemma
(2.3)(3)). Then has condition. So is
-reversible continuous.
E. Proposition 3.5
A retract of a -reversible quasi-continuous act is -reversible quasi continuous.
Proof: Assume that is a retract of a -reversible quasi-continuous act . In a similar
way to proposition 2.4, has condition.
Now, let and be two -reversible retracts of
with . Then and are two
-reversible retracts of and from
property of , so we have as a retract of .
But is a retract , so is a retract of .
Hence, has . Therefore, is
-reversible quasi-continuous.
The above concepts induce the next conditions for an act :
: Every -reversible subact of is -large in a fully invariant retract of
: Every -reversible subact of that is isomorphic to a retract of is a fully invariant retract of .
: If two -reversible retracts of have zero intersection, then their disjoint union is a fully invariant retract of .
F. Definition 3.6
An -act is called strongly -reversible
continuous if it satisfies the conditions
and .
F. Definition 3.7
An -act is called strongly -reversible
quasi-continuous if it satisfies the conditions
and .
G. Remarks and Examples 3.8
1. Each strongly -reversible (quasi-)
continuous act is -reversible (quasi-) continuous, but not conversely. For instance, consider
as a -act. Since is an
injective -act, then is injective,
and hence it is -reversible (quasi-) continuous.
But is not strongly -reversible (quasi-)
continuous since does not satisfy condition.
2. Each -reversible act is strongly
-reversible quasi-continuous. Note that as -act is strongly -reversible quasi-continuous, which is not -reversible.
3. In similar manner of lemma 3.3, if an
-act has condition, then has
condition.
4. By using [3], each strongly -reversible continuous act is strongly -reversible quasi-continuous. From that, it can be observed that as a -act is strong -reversible quasi-continuous,
which is not strongly -reversible (quasi)
continuous.
5. The following lemmas are useful because
they give us characterizations of strongly -reversible (quasi) continuous acts.
H. Lemma 3.9
An -act has a condition if and
only if has a condition and is a
-duo.
Proof: Assume that as an -act satisfies
the condition. So, obviously has a
condition. Take as a -reversible
retract of . By property of , since
so is a fully invariant retract of .
Hence is -duo. Conversely, let be a
-reversible subact of , such that , where
is a retract of . Therefore, by the
property, is a retract of . The -duo
property of brings as a fully invariant
subact. Therefore, has the
condition.
I. Lemma 3.10
An -act has condition if and
only if has a condition and is
-duo.
Proof: Evidently, by definitions.
By proposition 2.15., lemma 3.9. and lemma 3.1 have the next propositions:
J. Proposition 3.11
An -act is strongly -reversible
continuously if and only if is -reversible continuously and is a -duo.
K. Proposition 3.12
An -act is strongly -reversible
quasi-continuously if and only if is -reversible
quasi-continuously and is -duo.
L. Proposition 3.13
An -act is strongly -reversible
continuously if and only if satisfies the
condition and .
M. Proposition 3.14
An -act is strongly -reversible
continuously if and only if satisfies the
condition and .
An -act is strongly -reversible quasi-continuously if and only if satisfies the
condition and .
O. Proposition 3.16
An -act is strongly -reversible
quasi-continuously if and only if satisfies the
condition and .
P. Lemma 3.17
Every retract of the -duo act is -dou.
Proof: Let be a -duo act and be a retract of . Let be a -reversible retract of . Since is retract of , then is -reversible
retract of (by lemma 2.3 [3]). By -duo
property of , we have is a fully invariant
subact . Let be any homomorphism.
Then, can be extended to an endomorphism
of , such that where and
. Since is fully invariant of , thus
and so . Thus, is a fully
invariant subact of . Therefore, is -duo.
Q. Corollary 3.18
A retract of a strongly -reversible continuous act is strongly -reversible continuous.
Proof: From corollary 2.9, lemma 3.17 and
proposition 2.14.
R. Corollary 3.19
A retract of strongly - reversible quasi-continuous act is strongly -reversible quasi-continuous.
Proof: From proposition 2.12, lemma 2.17
and proposition 2.5.
IV. C
ONCLUSIONThe category of S-acts over monoids is a very important branch of mathematics and has various applications in computer science. Injective acts
over monoids was first constructed by
Berthiaume and was recently mentioned in many papers generalizing the injective S-acts. In this paper, we introduce and study new concepts in acts over monoids, which is a proper generalization of injective acts. We call an -act as (strongly) -reversible extending if every intersection reversible subact of is -large in
a (fully invariant) retract of . Many
characterizations and properties of -reversible extending acts are given. For example, an -act is strongly -reversible extending if and only if for each -reversible subact of , there is a
disjoint union , such that ,
with is fully invariant subact of and
is -large of . Also, we discuss the
strongly -reversible extending property as
closed under hereditary, retracts, and coproducts.
The relationship among strongly -reversible
extending S-acts and other known classes of acts over monoids.
A
CKNOWLEDGMENTThe first author would like to think
Mustansiriyah University
(https://uomustansiriyah.edu.iq), Baghdad, Iraq, for its support in the present work.
