FI-Extending Semimodule and Singularity

The main aim of this research is to present and to study several basic characteristics of the idea of FI-extending semimodules. The semimodule is said to be an FI-extending semimodule if each fully invariant subsemimodule of is essential in direct summand of . The behavior of the FI-extending semimodule with respect to direct summands as well as the direct sum is considered. In addition, the relationship between the singularity and FI-extending semimodule has been studied and investigated. Finally extending propertywhich is stronger than FI extending, that has some results related to FI-extending and singularity is also investigated.


Introduction
The originality of CS-modules is given by Von Neumann in 1930 [1]. In [2], Utumi in 1960 had identified and studied modules with a C1 condition in his research on the continuous and self-injective rings. The C1 condition is a common generalization of the semisimple and injective conditions. In [3], authors developed a CS condition which is another aspect of C1 condition . In the last years, extended modules theory has been came to ISSN: 0067-2904

Alhashemi and Alhossaini
Iraqi Journal of Science, 2022, Vol. 63, No. 3, pp: 1277-1284 1278 play a significant role and many researchers have been published and contributed to this hypothesis due to its interesting and widely available findings on expanding properties in the theoretical formulation of the module [4].
There are many generalization papers of extending property can be presented as follows: Wang and Wu [5] are studied the CLS-modules, as well as they provided a condition that makes the direct sum of CLS-modules to be CLS-module. In the other hand, in [6], authors introduced CLS-module ,and they developed the properties of y-closed submodules, by considering every y-closed submodule to be a direct summand. In [7], the concept of fully extending modules is introduced ,and proved that the class of fully extending modules is a proper subclass of the class of extending modules. Ungor and Halicioglu in [8] introduced a strongly extending module, and they investigated its properties as a particular extending module.
The idea of strongly extending modules is also defined by Atani, Hesari, and Khoramdel in [9] as a particular subclass of the class of extending modules, as well as they discussed some basic properties of this subclass of modules. The concept of generalized CS-module is defined by Zeng and Shaoguan in which homomorphic images of generalized CS-module, in addition the direct sum of semi-simple modules and singular modules are also generalized CS-module [10]. In [11], the concept of semi-extending modules, as a generalization of extending modules is studied by Ahmed and Abbas. In [12], authors defined the concept of FI-CS in which a direct sum of FI-CS modules is FI-CS, while the authors in [13] studied some conditions that applied to make the direct summand of FI-CS module is FI-CS module. Y¨ucel in [14] also introduced the generalized FI-CS module, and they demonstrated that the class of FI-CS modules is not closed under direct summands as well as they proved that it is closed under direct sums. Recently, a great attention in the field of semimodule has appeared via the study of many topics that are previously studied in the module converting to semimodules for more details see the following studies: Alhashemi and Alhossaini in [4] introduced and studied the extending semimodule over semiring, and they also studied some properties of the direct sum and direct summand of semimodule. In [15], the properties of singular and nonsingular semimodule are studied and investigated, as well as the relationship between singularity and extending semimodule is also proved in the same article. In this paper, we generalize the concept of extending semimodule by studying and introducing the FI-CS semimodule, and we also investigate some conditions that make the direct summand of FI-CS to be FI-CS, as well as we prove that the direct sum of FI-CS is always FI-CS. This paper is organized as follows: some preliminaries that are needed in this study are introduced in Section 2. The main contributions have been presented in Section 3 and Section 4, respectively. In Section 5 the concluding remarks of this work are given.

Preliminaries
In order to study and to investigate for FI-CS semimodule over a semiring, R denotes a commutative semiring with identity, and is a left R-semimodule. Definition 2.1 [16]: Let ( , +) be an additive abelian monoid with additive identity , then is called a left R-semimodule if there exists a scalar multiplication which is denoted by , such that Definition 2.2 [17]: A subset of an R-semimodule is called a subsemimodule of if for and , and and write ( ). Definition 2.3 [18]: A subsemimodule of is said to be fully invariant if for each R-endomorphism on (denoted ).
Definition 2.4 [19]: A nonzero R-subsemimodule of is called essential ,which is denoted by ( ) if 0 for every 0 . Definition 2.5 [20]: A subsemimodule of is defined by = { } is said to be singular subsemimodule of . If then is called singular. If , and is called nonsingular. Definition 2.6 [21]: The second singular subsemimodule of is that subsemimodule of , containing , such that is the singular subsemimodule of /Z( ). Definition 2.7 [22]: An R-semimodule is called a direct sum of subsemimodules 1 , 2 ,…, k of if each ϵ can be written uniquely as In this case each i is called a direct summand of (denoted by DS).

Definition 2.8[4]: An R-semimodule
is called extending (CS-semimodule) if every subsemimodule of is essential in a direct summand of . This is equivalent to following: Every closed subsemimodule of is a direct summand of . Definition 2.9 [23]: A subsemimodule of a semimodule is said to be closed if implies (denoted by ). Definition 2.10 [24]: A semimodule is said to be semisimple if it is a direct sum of its simple subsemimodule. Definition 2.11 [25]: A semimodule is said to be uniform if any subsemimodule of is essenti l Definition 2.12 [23]: If is an injective R-semimodule, and a minimal injective extension of the R-semimodule , then is said to be an injective hull of which is denoted by ( ). Definition 2.13 [26]

FI-CS Semimodule
The properties of the FI-CS semimodule are introduced and investigated in this section. It can be seen by analyzing the structure of FI-CS semimodule the following : There are many properties of fully invariant subsemimodules that are also useful. Next we will give some properties of a fully invariant submodule, these properties will be converted to subsemimodule. Lemma (3.

. FI-CS with Singularity
In this section, the relationship between the singularity and FI-CS semimodule for direct summands as well as the direct sum is studied and investigated. Proposition (4.  [15] there exists , such that hence . Since ) is nonsingular by [15] we have 2 ) = 0, therefore er , hence er = ', so er has no proper essential extension, then er is closed in 2  , for some nonsingular P of and P is -injective. Proof: It is easy to prove that hen = 0 or Z ( ) = . Now suppose that Z( ) , since Z ( ) is a fully invariant by [15], and is FI-CS, then there exists a DS subsemimodule of , such that Z ( ) , and = for some , so Z ( )= Z( Z( ), however Z( = Z( ) = Z( ). Therefore Z ( ) = 0 , hence is nonsingular), by Remark (4.6), Hom( , ) = 0, so that is Z ( )-injective . □ Proposition (4.8): For every FI-CS semimodule, Z 2 ( is FI-CS direct summand of F. Proof: Assume F is FI-CS, since Z 2 ( is a fully invariant in F, then by Proposition (3.6), Z 2 ( is FI-CS and there exists a direct summand F' of F such that . But is closed, hence is a direct summand of F. □

Conclusion
In general every module is semimodule, however the converse is not true. Thus most of the results which are achieved in FI-CS module they are also achieved in FI-CS semimodule. In addition, the additive cancellative property has been added as a condition in both Lemma (3.10) and Proposition (3.11) in order to obtain the results of this study. Likewise, some Propositions have been proven, for example, Lemma (3.11), Proposition (3. 12), Proposition (4.2), and Corollary (4.3). We have assumed that the existence of the injective hull, considering that this feature is always present in the module, which is not necessarily available in the semimodule. As a final result, it is clear that the purpose of this study was achieved.