On Fully Prime Radicals

In this paper we give a further study on fully prime submodules. For any fully prime submodules we define a product called $\am$-product. The further investigation of fully prime submodules in this work, i.e. the fully m-system and fully prime radicals, is related to this product. We show that the fully prime radical of any submodules can be characterize by the fully m-system. As a special case, the fully prime radical of a module $M$ is the intersection of all minimal fully prime submodules of $M$.


Introduction
Along with the process of generalization of rings to modules, some previous authors defined prime submodules as the generalization of prime ideals. By using some different approaches, there are some kinds of definition of prime submodules.
The definition of prime submodules introduced by Dauns [2] has been referred by many authors for their study of primeness in module theory. Investigations of prime radicals and localization of modules by Wisbauer [11] are also based on this definition.
In this work we refer to the definition of primeness of submodules based on the paper of Wijayanti and Wisbauer [8], in which they defined prime (sub)module using the * M -product and called it fully prime submodules. This * M -product was introduced earlier by Raggi et al. in their paper [7] and recently gives a possibility in a module to have such a "multiplication", such that we can adopt the definition of prime ideal to the module theory easier. The further study of the fully prime submodules started in paper Wijayanti [9]. The aim of this work is to complete the investigation of fully prime submodules, especially related to the m-system and radical.
Some previous authors also studied the prime submodules and prime radical submodules, for example Sanh [6], Lam [4] and Azizi [1]. We refer to Lam for defining fully m-systems and Sanh [6] for some properties of fully prime radicals of submodules.
In the next section we present some necessary and sufficient conditions of fully prime submodules in Proposition 2.2 and Proposition 2.5. Moreover, we show that there is a fully prime submodule which is not a prime submodule by giving a counter example in Example 2.3. To give some ideas of the fully primeness among another primeness, i.e. primeness in the sense of Dauns [2] and Sanh [5], we prove in Proposition 2.7, Proposition 2.9 and Lemma 2.10 that they not necessary coincide. Then we define a fully m-system and show that it is a complement of the fully prime submodule (see Proposition 2.13).
In the last section we define fully prime radicals and give some results. In Proposition 3.1 we characterize the fully prime radical of a submodule using the fully m-system. Moreover, in Proposition 3.9 we show that if the module is selfprojective, then the fully prime radical of factor module modulo its fully prime radical is equal to zero.
Throughout R denotes an associative ring with unit and the module should be a unital left module over the related ring. For our purpose, we write the homomorphism on the right side. A fully invariant submodule K in M is a submodule which satisfies (K)f ⊆ K for any endomorphism f of M . Naturally, for any R-module M , it is also a right S-module, where S = End R (M ) and the scalar multiplication is defined as µ : (m, f ) → (m)f .

Fully Prime Submodules
We begin this section by the definition of a product between two fully invariant submodules as we refer to [7] and [8]. For any fully invariant submodules K, L of M , consider the product We call the defined product in formulae (1) as the * M -product. Now we recall the definition of fully prime submodules in [8].
Definition 2.1. A fully invariant submodule P of M is called a fully prime submodule in M if for any fully invariant submodules K, L of M , A module M is called a fully prime module if its zero submodule is a fully prime submodule.
Moreover, we give a characterization of fully prime submodules as we can show in the following proposition. Proof. (a) ⇒ (b). Let m and k be elements in M where the fully invariant cyclic submodules < m > * M < k >⊆ P . Since P is fully prime, it implies < m >= mS ⊆ P or < k >= kS ⊆ P . Then m ∈ P or k ∈ P . Example 2.3. Let us give an example of a submodule which is not prime but fully prime. In Z-module Z 12 , we know that < 3 > is not a prime submodule. But < 3 > is a fully prime submodule, since for any proper submodules N, Now we recall some more properties according to Proposition 18 of Raggi et al. [7] which showed the relation between a fully prime submodule and the factor module. If a submodule N is fully prime, then the factor module M/N is also fully prime. But we need some property for the converse, as we can see below.
Based on Proposition 2.4, if a module M is self-projective, then a submodule P in M is fully prime if and only if M/P is fully prime. Let L and U be R-modules and recall the following definition: For detailed explanation of reject and cogenerator, the readers are suggested to refer to Wisbauer's book [10]. Moreover, we modify the result in 3.1 of [8] for a more general case. According to Proposition 2.5, if P is fully prime, then any M/P -cogenerated module is also K/P -cogenerated and vice versa.
Consider R as a left R-module and let I, J be ideals of R. Then I * R J = IJ. Since every ideal of R is a fully invariant R-submodule, we get the following special case: The following statements are equivalent for a two-sided ideal I : a. R/I is a prime ring.
b. I is a fully prime submodule in R.

c. I is a prime ideal.
A module M satisfies the ( * f i) condition if for any non-zero fully invari- In general prime modules in the sense of Dauns [2] need not to be fully prime. Furthermore, in any self-projective module, if its submodule is prime, then it is not necessary fully prime. For the following relationship we generalize Proposition 3.4 of [8] as follows.
Proposition 2.7. For a self-projective R-module M with ( * f i) and for any fully invariant submodules K of M , the following statements are equivalent : Notice that for any ring R, End R (R) ≃ R and as a left R-module, R satisfies ( * f i) and is fi-retractable. If M = R, Corollary 3.5 and Proposition 3.4 of [8] show that primeness and fully primeness of R coincide.
For any fully invariant submodule K of M we denote Now we recall the definition of prime submodule in the sense of Sanh et al. [5], and we called it N-prime, as follows. We refer to Theorem 1.2 of Sahn et. al. [5] to give a characterization of an N-prime submodule and consider that the prime notion in the sense of Sanh (N-prime) and endo-prime notion in the sense of Haghany-Vedadi [3] coinside. As an immediate consequence we extend Corollary 1.6 of [3] as follows.
As a consequence of Proposition 2.7 and Proposition 2.9 we have the following property.
Lemma 2.10 gives a sufficient condition of a fully prime module to be N-prime. Now we refer to Theorem 1.2 of Sahn et. al. [5] to prove the following property. Related to fully prime submodules, we observe now the notion below.
Definition 2.12. Let X be a non empty subset of a module M where 0 ∈ X. X is called a fully m-system if for any x, y ∈ X, < x > * M < y > ∩X = ∅.
As it is already known, there is a similar m-system notion in rings and it is a complement set of a prime ideal (see for example Lam [4]). In modules, we prove that the fully m-system is also a complement set of a fully prime submodule. Proof. (⇒). Denote X = M \ P . Take any x, y ∈ X, then x, y ∈ P . By Proposition 2.2 (b), < x > * M < y > ⊆ P , hence < x > * M < y > ∩X = ∅. Hence X = M \ P is a fully m-system. (⇐). Take any x, y ∈ P , then x, y ∈ M \ P and < x > * M < y > ⊆ P . Hence P is fully prime.
Next we show that any maximal fully invariant submodule is also a fully prime submodule.
Proposition 2.14. Let X be a fully m-system in M and P be a maximal fully invariant submodule in M where P ∩ X = ∅. Then P is a fully prime submodule in M .
Proof. Suppose x ∈ P and y ∈ P , but < x > * M < y >⊆ P . Then there exist x 1 , x 2 ∈ X such that x 1 ∈ P + < x > and x 2 ∈ P + < y >. Consider that This is a contradiction with the fact that X is a fully m-system.
Moreover, we also show that any fully prime submodule contains a minimal fully prime submodule. It is clear that J = ∅, since P ∈ J. By Zorn's Lemma, J has a minimal element or equivalently, every nonempty chain in J has a lower bound in J. Consider a nonempty chain G ⊆ J. We construct a set Q = K∈G K. It is clear that Q is a fully invariant submodule in M and Q ⊆ P . We want to show that Q is a fully prime submodule in M . Take any two fully invariant submodules X and Y in M where X * M Y ⊆ Q but Y Q. We prove that X ⊆ Q. Take any y ∈ Y \ Q. Then there exists K ′ ∈ G such that y ∈ K ′ . Since K ′ is a fully prime submodule in M , from X * M Y ⊆ Q ⊆ K ′ implies X ⊆ K ′ . Then take any L ∈ G. Since G is a chain in J, Thus X ⊆ L for all L ∈ G. Then X ⊆ Q, and we prove that Q is a fully prime submodule in M . Since Q ⊆ P , Q ∈ J and is a lower bound of G. It is proved that any nonempty chain in J has a lower bound in J. Based on Zorn's Lemma there exists a fully prime submodule P * ∈ J which is minimal in J. Thus the fully prime submodule P contains the minimal fully prime submodule P * .
Let N and K be fully invariant submodules of M where K ⊆ N . We consider then the sets Hom R (M, K) and Hom R (N, K). For any f ∈ Hom R (M, K), it induces a homomorphismf = f | N ∈ Hom R (N, K). It is understood that there exists an injective function from Hom R (M, K) to Hom R (N, K) which maps any f ∈ Hom R (M, K) tof = f | N ∈ Hom R (N, K). Hence, Hom R (M, K) ⊆ Hom R (N, K).
Proposition 2.17. Let N and P be a fully invariant submodules of M . If P is fully prime, then N ∩ P is a fully prime submodule in N .
Proof. Take any two fully invariant submodules K and L in N , where L * N K ⊆ N ∩ P . Then L * N K ⊆ P and LHom R (N, K) ⊆ P . Since Hom R (M, K) ⊆ Hom R (N, K), LHom R (M, K) ⊆ LHom R (N, K). Moreover L * M K ⊆ L * N K and hence L * M K ⊆ P . Then L ⊆ P or K ⊆ P because P is fully prime. But K ⊆ N and L ⊆ N , so we have K ⊆ N ∩ P or L ⊆ N ∩ P as well.
We recall Lemma 17 of Raggi et al. [7] below. As a consequence, we have the following property. Conversely, take any a ∈ Rad f p M (0), then there exists a minimal fully prime submodule P ∈ Spec f p (M ) such that a ∈ P . Consequently a ∈ P ′ ∈J P ′ , and we prove  Proof. Take any X ∈ λ∈Λ ν f p (L λ ) and then X ∈ ν f p (L λ ) for all λ ∈ Λ. It implies that X is a fully prime submodule which contains L λ for all λ ∈ Λ. Hence X also contains the sum λ∈Λ L λ and X ∈ ν f p ( λ∈Λ L λ ). Conversely, now take Y ∈ ν f p ( λ∈Λ L λ ). It means Y is a fully prime submodule and λ∈Λ L λ ⊆ Y . Then for all λ ∈ Λ we obtain L ∈ ν f p (L λ ) since L λ ⊆ Y . Hence Y ∈ ν f p ( λ∈Λ L λ ). Proof. Let P ∈ ν f p (N ) ∪ ν f p (L). We have P ∈ ν f p (N ) or P ∈ ν f p (L). As a consequence we obtain N ⊆ P or L ⊆ P . Hence, N ∩L ⊆ N ⊆ P or N ∩L ⊆ L ⊆ P . Thus we conclude that N ∩ L ⊆ P , or in other words P ∈ ν f p (N ∩ L).
Let I be an ideal in S. Then we denote

Concluding Remarks
Further work on the properties of fully prime radicals of submodules can be carried out. For example, we can define a fully multiplication module and then investigate the properties of fully prime radicals on the fully multiplication module. Moreover, we can bring this work to define fully prime localizations and then investigate the fully prime localization on fully multiplication modules.