# On Nearly Prime Submodules of Unitary Modules.

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 R-modules. For a right R-module M, we denote S = [End.sub.R](M) for its endomorphism ring. A submodule X of M is called a fully invariant submodule of M if, for any f [member of] S, we have f(X) [subset] X. A right R-module M 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 X of a right ^-module M is a proper fully invariant submodule of M with the property that, for any ideal I of S = [End.sub.R](M) and any fully invariant submodule U of M, I(U) [subset] X implies I(M) [subset] X or U [subset] X. 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 X is invariant under [phi]S instead of requiring the submodule X 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.

General background materials can be found in [9-12].

2. Main Results

We introduce the definition of a nearly prime submodule by a weaker condition that X is invariant under [phi]S, instead of requiring the submodule X to be fully invariant.

Definition 1. A proper submodule X of a right R-module M is called a nearly prime submodule if, for any [phi] [member of] S and for any m [member of] M, if [phi]S(m) [subset] X and [phi]S(X) [subset] X, then either m [member of] X or [phi](M) [subset] X Particularly, a proper right ideal P of R is a nearly prime right ideal if for a, b [member of] R such that aRb [subset] P and aRP [subset] P, then either a [member of] P or b [member of] P.

From these definitions, any prime submodule of a right R-module M is nearly prime.

In the following theorem and its corollary, we can see that a proper right ideal P of R is nearly prime if for any right ideals A, B [subset] R such that AP [subset] P and AB [subset] P, then either A [subset] P or B [subset] P. Note that Koh  gave this definition and used the terminology prime right ideals.

Theorem 2. Let X be a proper submodule of M. The following conditions are equivalent.

(1) X is a nearly prime submodule of M.

(2) For any right ideal I of S, any submodule U of M, if I(U) [subset] X and I(X) [subset] X, then either I(M) [subset] X or U [subset] X.

(3) For any [phi] [member of] S and fully invariant submodule U of M, if [phi](U) [subset] X and [phi]S(X) [subset] X, then either [phi](M) [subset] X or U [subset] X.

Corollary 3. Let I be a proper right ideal of a ring R. The following conditions are equivalent.

(1) I is a nearly prime right ideal of R.

(2) For any right ideal A, B of R, if AB [subset] I and AI [subset] I, then either A [subset] I or B [subset] l.

(3) For any a [member of] R and any ideal B of R, if aB [subset] I and aRI [subset] I, then either a [member of] I or B [subset] I.

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 I of S = [End.sub.R](M), any fully invariant submodule U of M, if I(U) [subset] X, then either I(M) [subset] X or U [subset] X. By our definition, any prime submodule of M is nearly prime.

(2) Also by Sanh et al. , if X is a maximal fully invariant submodule of M, then X is prime. We now show that any maximal submodule X of M is nearly prime. In fact, let [phi](U) [subset] X, where U is a submodule of M and [phi] [member of] S with [phi]S(X) [subset] X. Suppose that U [subset or not equal to] X. Then there is an u [member of] U such that X + uR = M. This follows that [phi](M) = [phi](X) + [phi](uR) = [phi](X) + [phi](u)R [subset] X since [phi](U) [subset] X. This shows that X is nearly prime. Note that, in general, a maximal submodule of a right R-module M does not need to be fully invariant. Therefore the class of nearly prime submodules of a given right R-module M 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 D be a division ring and let R be the following subring of [M.sub.3] (D) :

[mathematical expression not reproducible]. (1)

Let P [subset] R be the right ideal consisting of matrices in R whose first row is zero, i.e., [mathematical expression not reproducible]. Now, we assume that aRb [subset] P and aRP [subset] P for arbitrary a, b [member of] R. Since aRP [subset] P, we get [a.sub.11] + [a.sub.12] = 0 and [a.sub.11] + [a.sub.13] = 0. And hence

[mathematical expression not reproducible]. (2)

This would imply that either a [member of] P or b [member of] P, so P is a nearly prime right ideal of R.

One useful generalization of nearly prime submodule is obtained by replacing the condition "[phi]S(m) [subset] X and [phi]S(X) [subset] X"; we called it nearly strongly prime submodule.

Definition 5 (see ). A proper submodule X of a right R-module M is called a nearly strongly prime submodule if, for any [phi] [member of] S and m [member of] M, if [phi](m) [subset] X and [phi](X) [subset] X, then either m [member of] X or [phi](M) [subset] X.

To see the relationship between a nearly prime submodule and nearly strongly prime submodule, we will need the following terminology.

Definition 6. A submodule X of a right R-module M is called to have insertion factor property (briefly, an IFP-submodule), if for all [phi][member of] S = [End.sub.R](M) and m [member of] M, [phi](m) [member of] X, then [phi]S(m) [subset or equal to] X.

Then the relationship between a nearly prime submodule and nearly strongly prime submodule is as follows.

Proposition 7. Let M be a right R-module and X a submodule of M. If X is a nearly strongly prime submodule of M, then X is a nearly prime submodule of M.

Proof. The proof is immediate.

Proposition 8. Let M be a right R-module and X a submodule of M. If X is a nearly prime submodule of M and has insertion factor property, then X is a nearly strongly prime submodule of M.

Proof. Let p [member of] S = [End.sub.R](M) and m [member of] M such that [phi](m) [member of] X and [phi](X) [subset or equal to] X. Since X is an IFP-submodule of M, [phi]S(m) [subset or equal to] X. Let x [member of] X. Since [phi](X) [subset or equal to] X, p(x) [member of] X. But X is an IFP-submodule of M; we have [phi]S(x) [subset or equal to] X. So [phi]S(X) [subset or equal to] X. Since X is a nearly prime submodule of M, m [member of] M or [phi](M) [subset or equal to] X. Therefore X is a nearly strongly prime submodule of M.

Proposition 9. If X is a nearly prime submodule of M, then X contains a minimal nearly prime submodule of M.

Proof. Let F be the set of all nearly prime submodules of M which are contained in X. Since X [member of] 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 [subset] F has a lower bound Q in F. Put [mathematical expression not reproducible]; then [phi]S(Q) [subset] Q for any p [member of] S. We will show that Q is a nearly prime submodule of M and Q [subset] X. Suppose that p [member of] S and m [member of] M/Q such that [phi]S(m) [member of] Q. Since [mathematical expression not reproducible], there exists N [member of] G with m [not member of] N. By the nearly primeness of N, we have [phi](M) [subset] N. For any U [member of] G, either F [subset] N or N [subset] U. If U [subset] X, we see that m [not member of] U, which implies that [phi](M) [subset] U by the nearly primeness of U. If N [subset] U, we have [phi](M) [subset] N [subset] U. Thus [phi](M) [subset] U for any U [member of] G. Hence [phi](M) [subset] Q, proving that Q is nearly prime in M. It is clear that Q [subset] X. Therefore, Q is a lower bound for G. Again by Zorn's Lemma, there exists a nearly prime [P.sup.*] which is minimal among the nearly prime submodules in F. Since any nearly prime submodule contained in [P.sup.*] is in F, we conclude that [P.sup.*] is a minimal nearly prime submodule of M.

Let X be a submodule of M. Then the set [I.sub.X] = {f [member of] S | f(M) [subset] X} is a right ideal of S. In the following theorem, we consider the relation between X and [I.sub.X].

Theorem 10. Let M be a right R-module which is a self-generator and X be a submodule of M. If X is a nearly prime submodule, then [I.sub.X] = {f [member of] S | f(M) [subset] X} is a nearly prime right ideal of S. Conversely, if [I.sub.X] is a nearly prime right ideal of S, then X is a nearly prime submodule of M.

Proof. Since [phi]S[I.sub.x] [subset] [I.sub.X], we have [phi]S[I.sub.X](M) [subset] [I.sub.X](M), and since M is a self-generator, we have [I.sub.X](M) = X and hence [phi]S(X) [subset] X. Take any [psi] [member of] S. If [phi][S.sub.[psi]] [subset] [I.sub.X], then [phi][S.sub.[PSI]])(M) [subset] X. This follows that [phi][S.sub.[PSI]](m) [subset] X, for all m [member of] M. Hence [phi](M) [subset] X or [psi](m) [member of] X, showing that [phi] [member of] [I.sub.X] or [psi] [member of] [I.sub.X].

Conversely, suppose that [I.sub.X] is nearly prime. Let [phi]S(m) [subset] X and [phi]S(X) [subset] X. We have [phi]S(m)R [subset] X and it would imply that [phi]S(mR) [subset] X. Since mR = [[summation].sub.g[member of]A](M) for some subset A of S,

[mathematical expression not reproducible]. (3)

This would imply that [phi]Sg [subset] [I.sub.X]. It follows from the hypothesis that [phi] [member of] [I.sub.X] or g [member of] [I.sub.X]. This shows that [phi](M) [subset] X or g(M) [subset] X for all g [member of] A. Hence [phi](M) [subset] X or m [member of] X, proving our theorem.

Recall from  that a module X is called M-generated if there is an epimorphism [M.sub.(I)] [right arrow] N for some index set I. If I is finite, then N is called finitely M-generated. In particular, a module N is called M-cyclic if there is an epimorphism from M [right arrow] N.

Lemma 11. Let M be a quasi-projective module and X be an M-cyclic submodule of M. Then [I.sub.X] is a principal right ideal of S.

Proof. Since X is M-cyclic, there exists an epimorphism [phi] : M [right arrow] X such that X = [phi](M). It follows that [phi]S [subset] [I.sub.X]. By the quasi-projective of M, for any f [member of] [I.sub.X], we can find a [psi] [member of] S such that f = [phi][psi] proving that [I.sub.X] = [phi]S. Hence [I.sub.X] is a principal right ideal of S.

Proposition 12. Let M be a right R-module and X be a submodule of M. If f is injective for all 0 [not member of] f [member of] [End.sub.R](M/X), then X is a nearly strongly prime submodule of M.

Proof. Suppose that f is injective for all 0 [not equal to] f [member of] [End.sub.R](M/X). Let [phi] [member of] S = [End.sub.R](M) and m [member of] M such that [phi](m) [member of] X and [phi](X) [subset or equal to] X. Assume that [psi](M) [not member of] X. Then there exists m [member of] M such that [phi](m) [not member of] X. Define f : M/X [right arrow] M/X by f(x + X) = [phi](x) + X for all x [member of] X. Let [x.sub.1], [x.sub.2] [member of] X such that [x.sub.1] + X = [x.sub.2] + X. We have [x.sub.1] - [x.sub.2] [member of] X but [phi](X) [subset or equal to] X, [phi]([x.sub.1] - [x.sub.2]) [member of] X, and hence [phi]([x.sub.1]) + X = [phi]([x.sub.2]) + X. So f is well-defined and it is clear that f is an R-homomorphism. Since f(m + X) = [phi](m) + X [not equal to] X, f [not equal to] 0. By assumption, f is injective. Then f(m + X) = [phi](m) + X = X but f is an injective, m + X = X, and hence m [member of] X. Therefore X is a nearly strongly prime submodule of M.

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 M be a finitely generated right R-module. Then M is a Noetherian module if and only if every nearly prime submodule of M 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 R is generated by one element if and only if every prime right ideal of R is generated by one element.

Theorem 15. Let M be a quasi-projective, finitely generated right R-module which is a self-generator. If all of nearly prime submodules of M are M-cyclic submodules, then every ideal in S is principal.

Proof. Let P be a prime right ideal of S and X = P(M). Since M is finitely generated and quasi-projective, it follows from [18.4] that P = [I.sub.X] and, therefore, X is a nearly prime submodule of M by Theorem 10. Moreover, by hypothesis and Lemma 11, we can see that P is a principal right ideal of S. It follows from Theorem 14 that every right ideal of R is generated by one element.

Corollary 16. Let M be a quasi-projective, finitely generated right R-module which is a self-generator. If all of nearly prime submodules of M are M-cyclic submodules, then every submodule of M is M-cyclic.

Proof. The proof is immediate.

For M = [R.sub.R], the next corollary follows consequently.

Corollary 17. A ring R 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.

Data Availability

No data were used to support this study.

https://doi.org/10.1155/2018/7202590

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research is supported by Rachadapisek Sompot Fund for Postdoctoral Fellowship, Chulalongkorn University.

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Nguyen Dang Hoa Nghiem, (1) Samruam Baupradist (iD), (1,2) and Ronnason Chinram (iD) (3)

(1) Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

(2) Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand

(3) Algebra and Applications Research Unit, Department of Mathematics and Statistics, Faculty of Science,

Prince of Songkla University, Hat Yai, Songkhla 90110, Thailand

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