On Quasi-Small Prime Submodules

__________________________________________ Abstract Let be a commutative ring with identity , and be a unitary (left) R-module. A proper submodule of is said to be quasi- small prime submodule , if whenever with 〈 〉 and , then either or . In this paper ,we give a comprehensive study of quasi- small prime submodules.


Introduction:
Throughout this research, we consider is a commutative ring with identity and is a unity R-module. A proper submodule of is said to be prime if whenever implies either [ ] , where [ ] { }, see [1]. As a generalization of the concept of prime modules and prime submodules, Mahmood, L.S. in 2012, [2] introduced the concepts of small prime modules and small prime submodules. In fact, the concepts of small semiprime modules and small semiprime submodules are also introduced in 2021 by Haider A. Ramadhan, Nuhad S. Al. Mothafar, [3]. A submodule of is called small (notational , if for all submodules of implies [4]. A quasi-prime submodules is introduced and studied in 1999 by Abdul-Razak, M. H. [5] as a generalization of a prime submodules. The proper submodule of an R-module is called a quasi-prime if whenever , where implies that either or . As a generalization of these concepts Wisam, A. Ali. and Al-Mothafar, N.S. [6] were introduced the concept of quasi-small prime modules in 2021. They call a quasi-small prime module if is prime ideal for all non-zero small submodule of .
The main purpose of this paper is to study the basic properties of quasi-small prime submodules.

Quasi-Small Prime Submodules
In this section we introduce and study the concept of a quasi-small prime submodule. . 3. It is clear that every prime submodule is quasi-small prime. Next example shows that the converse of (3) is not true in general. Example: Consider as a Z-module, ̅ . is a quasi-small prime submodule , since is a small prime submodule [2], hence it is a quasi-small prime submodule by (2), which is not prime, since ̅ ̅ , but ̅ ̅ and [ ̅ ] 4. Consider as a Z-module, let n is a positive integer. If n is a prime number, then the submodule is a quasi-small prime submodule, since if is prime number, then is a quasi-prime submodule, [5]. Hence by (1), is a quasi-small prime submodule. 5. Every quasi-small prime submodule is a small semiprime submodule, where a proper submodule of a module is called a small semiprime submodule of if and only if whenever with 〈 〉 and implies [3]. Proof: Let be a quasi-small prime submodule of an R-module . If for and with 〈 〉 , then by definition (2.1). Hence, is a small semiprime submodule of . We give the following example to show that the converse of (5) is not true in general. Example: is a small semiprime submodule of , [3], but is not a quasi-small prime submodule of Z by (4). 6. If is an R-module in which every cyclic submodule is small, then every quasi-small prime submodule of is a small prime submodule. 7. If is a hollow R-module, then every quasi-small prime submodule of is a quasi-prime submodule, where an R-module is called a hollow module if every proper submodule of is small, [4]. 8 [11]. It follows that [ ] is a quasi-small prime ideal of . Proposition (2.21): Let be a faithful finitely generated multiplication R-module and be a proper submodule of , the following statements are equivalent: 1. is a quasi-small prime submodule of .

More Properties a bout Quasi-Small Prime Submodules.
Recall that an R-module is called a quasi-small prime modules, if and only if is a prime ideal of R for all non-zero small submodule of , [6]. We know that an R-module is a prime (quasi-prime) if and only if 〈 〉 is a prime (quasi-prime) submodule of , [ (1) By proposition(3.1).

Proposition (3.4):
If is a quasi-small prime R-module, then is a quasi-small prime submodule of for every ideal of . Proof: If is a quasi-small prime R-module, then is a quasi-small prime submodule of by proposition (3.1). But [ ] , [10, p.16]. Thus, is a quasi-small prime submodule of , by proposition (2.15).