On Nearly Prime Submodules of Unitary Modules

The aim of this paper is to introduce the notion of nearly prime submodules as a generalization of prime submodules.We investigate some of their basic properties and point out the similarities between these submodules and the prime submodules.We also indicate some applications of nearly prime submodules. These applications show how nearly prime submodules control the structure of modules and they recover earlier relative theorems.


Introduction and Preliminary
Modules over associative rings play important roles in the investigation of ring constructions (see [1,2]).Modules are very important and have been actively investigated (see, for example, [3][4][5][6][7]).Throughout this paper, all rings are associative with identity and all modules are unitary right modules.For a right -module , we denote  =   () for its endomorphism ring.A submodule  of  is called a fully invariant submodule of  if, for any  ∈ , we have () ⊂ .A right -module  is called a self-generator if it generates all submodules.
In 2008, Sanh et al. [8] introduced the new notion of prime and semiprime submodules.Following that, a prime submodule  of a right -module  is a proper fully invariant submodule of  with the property that, for any ideal  of  =   () and any fully invariant submodule  of , () ⊂  implies () ⊂  or  ⊂ .We can say that this new approach is nontrivial, creative, and well-posed.We already got many results using those new notions that are unparalleled.As an extension of this work, we generalize the notion of a prime submodule.
Many people generalize the notion of a prime submodule.To do that, there are several ways but we put our attention to replace a weaker condition that  is invariant under  instead of requiring the submodule  to be fully invariant, and we called it nearly prime submodule.Using this new definition, we proved many meaningful properties of nearly prime submodules which are similar to that of prime submodules and also prime ideals.

Main Results
We introduce the definition of a nearly prime submodule by a weaker condition that  is invariant under , instead of requiring the submodule  to be fully invariant.From these definitions, any prime submodule of a right -module  is nearly prime.
In the following theorem and its corollary, we can see that a proper right ideal  of  is nearly prime if for any right ideals ,  ⊂  such that  ⊂  and  ⊂ , then either  ⊂  or  ⊂ .Note that Koh [13] gave this definition and used the terminology prime right ideals.Theorem 2. Let  be a proper submodule of . e following conditions are equivalent.
(1)  is a nearly prime submodule of .
( (1)  is a nearly prime right ideal of .
(3) For any  ∈  and any ideal  of , if  ⊂  and  ⊂ , then either  ∈  or  ⊂ .
Next, we give some examples and remark of nearly prime submodules and nearly prime right ideals; we maintain the notion and terminology as in [8].
Example .(1) Following Sanh et al. [8], a fully invariant is a prime submodule if, for any ideal  of  =   (), any fully invariant submodule  of , if () ⊂ , then either () ⊂  or  ⊂ .By our definition, any prime submodule of  is nearly prime.
(2) Also by Sanh et al. [8], if  is a maximal fully invariant submodule of , then  is prime.We now show that any maximal submodule  of  is nearly prime.In fact, let () ⊂ , where  is a submodule of  and  ∈  with () ⊂ .Suppose that  ̸ ⊂ .Then there is an  ∈  such that  +  = .This follows that () = () + () = () + () ⊂  since () ⊂ .This shows that  is nearly prime.Note that, in general, a maximal submodule of a right -module  does not need to be fully invariant.Therefore the class of nearly prime submodules of a given right -module  is larger than that of prime submodules.As a consequence, every maximal right ideal is a nearly prime right ideal.
(3) The following example is due to Reyes [14].Let  be a division ring and let  be the following subring of M 3 () : Let  ⊂  be the right ideal consisting of matrices in  whose first row is zero, i.e.,  := ( 0 0 0 0  0 0 0 This would imply that either  ∈  or  ∈ , so  is a nearly prime right ideal of . One useful generalization of nearly prime submodule is obtained by replacing the condition "() ⊂  and () ⊂ "; we called it nearly strongly prime submodule.Definition (see [15]).A proper submodule  of a right module  is called a nearly strongly prime submodule if, for any  ∈  and  ∈ , if () ⊂  and () ⊂ , then either  ∈  or () ⊂ .
To see the relationship between a nearly prime submodule and nearly strongly prime submodule, we will need the following terminology.
Then the relationship between a nearly prime submodule and nearly strongly prime submodule is as follows.
Proposition 7. Let  be a right -module and  a submodule of .If  is a nearly strongly prime submodule of , then  is a nearly prime submodule of .
Proof.The proof is immediate.Proposition 8. Let  be a right -module and  a submodule of .If  is a nearly prime submodule of  and has insertion factor property, then  is a nearly strongly prime submodule of .
Proof.Let F be the set of all nearly prime submodules of  which are contained in .Since  ∈ F, F is nonempty.By Zorn's Lemma, F has a minimal element with respect to the inclusion operation provided; we show that any nonempty chain G ⊂ F has a lower bound  in F. Put  = ⋂ ∈G ; then () ⊂  for any  ∈ .We will show that  is a nearly prime submodule of  and  ⊂ .Suppose that  ∈  and  ∈ / such that () ∈ .Since  ∉  = ⋂ ∈G , there exists  ∈ G with  ∉ .By the nearly primeness of , we have () ⊂ .For any  ∈ G, either  ⊂  or  ⊂ .If  ⊂ , we see that  ∉ , which implies that () ⊂  by the nearly primeness of .If  ⊂ , we have () ⊂  ⊂ .Thus () ⊂  for any  ∈ G. Hence () ⊂ , proving that  is nearly prime in .It is clear that  ⊂ .Therefore,  is a lower bound for G. Again by Zorn's Lemma, there exists a nearly prime  * which is minimal among the nearly prime submodules in F. Since any nearly prime submodule contained in  * is in F, we conclude that  * is a minimal nearly prime submodule of .
Let  be a submodule of .Then the set   = { ∈  | () ⊂ } is a right ideal of .In the following theorem, we consider the relation between  and   .
Theorem 10.Let  be a right -module which is a selfgenerator and  be a submodule of .If  is a nearly prime submodule, then   = { ∈  | () ⊂ } is a nearly prime right ideal of .Conversely, if   is a nearly prime right ideal of , then  is a nearly prime submodule of .
Recall from [16] that a module  is called -generated if there is an epimorphism  () →  for some index set .If  is finite, then  is called finitely -generated.In particular, a module  is called -cyclic if there is an epimorphism from  → .Lemma 11.Let  be a quasi-projective module and  be an -cyclic submodule of .en   is a principal right ideal of .
Proof.Since  is -cyclic, there exists an epimorphism  :  →  such that  = ().It follows that  ⊂   .By the quasi-projective of , for any  ∈   , we can find a  ∈  such that  = , proving that   = .Hence   is a principal right ideal of .Proposition 12. Let  be a right -module and  be a submodule of .If  is injective for all 0 ̸ =  ∈   (/), then  is a nearly strongly prime submodule of .

Applications of Nearly Prime Submodules
The following theorem shows how nearly prime submodules control the structure of a finitely generated module.Moreover, this theorem can be considered as a generalization of Cohen's theorem, a famous theorem in commutative algebra.
Theorem 13.Let  be a finitely generated right -module.
en  is a Noetherian module if and only if every nearly prime submodule of  is finitely generated.
In this section, we will show other applications of nearly prime submodules.The following results had appeared in [13] and we propose them here to use later on.

Theorem 14. Every right ideal of 𝑅 is generated by one element if and only if every prime right ideal of 𝑅 is generated by one element.
Theorem 15.Let  be a quasi-projective, finitely generated right -module which is a self-generator.If all of nearly prime submodules of  are -cyclic submodules, then every ideal in  is principal.
Proof.Let  be a prime right ideal of  and  = ().Since  is finitely generated and quasi-projective, it follows from [12][18.4] that  =   and, therefore,  is a nearly prime submodule of  by Theorem 10.Moreover, by hypothesis and Lemma 11, we can see that  is a principal right ideal of .It follows from Theorem 14 that every right ideal of  is generated by one element.

Corollary 16.
Let  be a quasi-projective, finitely generated right -module which is a self-generator.If all of nearly prime submodules of  are -cyclic submodules, then every submodule of  is -cyclic.
Proof.The proof is immediate.
For  =   , the next corollary follows consequently.
Corollary 17.A ring  is a principal right ideal ring if and only if all of its nearly prime right ideals are principal.
Particularly, the following corollary is very useful in commutative algebra.
Particularly, a proper right ideal  of  is a nearly prime right ideal if for ,  ∈  such that  ⊂  and  ⊂ , then either  ∈  or  ∈ .
Definition .A proper submodule  of a right -module  is called a nearly prime submodule if, for any  ∈  and for any  ∈ , if () ⊂  and () ⊂ , then either  ∈  or () ⊂ .