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.
1. 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, [3–7]). 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.  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.
2. 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.
Definition 1. A proper submodule of a right -module is called a nearly prime submodule if, for any and for any , if and , then either or Particularly, a proper right ideal of is a nearly prime right ideal if for such that and , then either or
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  gave this definition and used the terminology prime right ideals.
Theorem 2. Let be a proper submodule of The following conditions are equivalent.(1) is a nearly prime submodule of (2)For any right ideal of , any submodule of , if and , then either or (3)For any and fully invariant submodule of , if and , then either or
Corollary 3. Let be a proper right ideal of a ring The following conditions are equivalent. (1) is a nearly prime right ideal of (2)For any right ideal of , if and , then either or (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 .
Example 4. (1) Following Sanh et al. , 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. , 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 . Let be a division ring and let be the following subring of Let be the right ideal consisting of matrices in whose first row is zero, i.e., Now, we assume that and for arbitrary Since , we get and And hence 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 5 (see ). 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.
Definition 6. A submodule of a right -module is called to have insertion factor property (briefly, an IFP-submodule), if for all and , , then .
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 and such that and . Since is an IFP-submodule of , Let . Since , . But is an IFP-submodule of ; we have . So . Since is a nearly prime submodule of , or . Therefore is a nearly strongly prime submodule of .
Proposition 9. If is a nearly prime submodule of , then contains a minimal nearly prime submodule of
Proof. Let be the set of all nearly prime submodules of which are contained in . Since , is nonempty. By Zorn’s Lemma, has a minimal element with respect to the inclusion operation provided; we show that any nonempty chain has a lower bound in Put ; then for any We will show that is a nearly prime submodule of and . Suppose that and such that Since , there exists with By the nearly primeness of , we have For any , either or If , we see that , which implies that by the nearly primeness of If , we have Thus for any Hence , proving that is nearly prime in . It is clear that Therefore, is a lower bound for Again by Zorn’s Lemma, there exists a nearly prime which is minimal among the nearly prime submodules in Since any nearly prime submodule contained in is in , 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 self-generator 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 .
Proof. Since , we have , and since is a self-generator, we have and hence . Take any . If , then . This follows that , for all . Hence or , showing that or .
Conversely, suppose that is nearly prime. Let and . We have and it would imply that . Since for some subset of ,This would imply that . It follows from the hypothesis that or This shows that or for all Hence or , proving our theorem.
Recall from  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 . Then 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 , then is a nearly strongly prime submodule of .
Proof. Suppose that is injective for all . Let and such that and . Assume that . Then there exists such that . Define by for all Let such that . We have but , , and hence . So is well-defined and it is clear that is an -homomorphism. Since , . By assumption, is injective. Then but is an injective, , and hence . Therefore is a nearly strongly prime submodule of .
3. 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. Then 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  and we propose them here to use later on.
Theorem 14. Every right ideal of is generated by one element if and only if every prime right ideal of 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  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.
Corollary 18. An integral domain is a principal ideal domain (PID) if every prime ideal is principal.
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research is supported by Rachadapisek Sompot Fund for Postdoctoral Fellowship, Chulalongkorn University.
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