SOME PROPERTIES OF LEFT WEAKLY JOINTLY PRIME (R, S)-SUBMODULES

Let R and S be commutative rings and M be an (R,S)-module. A proper (R,S)-submodule P of M is called a left weakly jointly prime if for each elements a and b in R and (R,S)-submodule K of M with abKS ⊆ P implies either aKS ⊆ P or bKS ⊆ P . In this paper, we present some properties of left weakly jointly prime (R,S)-submodule. We give some necessary and sufficient conditions of left weakly jointly prime (R,S)-submodules. Moreover, we present that every left weakly jointly prime (R,S)-submodule contains a minimal left weakly jointly prime (R,S)-submodule. At the end of this paper, we show that in left multiplication (R,S)-module, every left weakly jointly prime (R,S)-submodule is equal to jointly prime (R,S)-submodules.


INTRODUCTION
Throughout this paper, ring R and ring S will denote commutative rings, and R-module M means an Abelian group under addition. The concept of the R-module has been studied in depth in Adkins [1].
An R-module has been generalized into an (R, S)-bimodule. When R and S are arbitrary rings, Khumprapussorn et al. [7] have generalized (R, S)-bimodule into (R, S)-module. An (R, S)-module has an (R, S)-bimodule structure when both rings R and S have central idempotent elements. When R and S are rings with identity, we have an (R, S)-module is also an (R, S)-bimodule.
Moreover, an (R, S)-submodule of an (R, S)-module M is a subgroup N of M such that rns ∈ N for all r ∈ R, n ∈ N , and s ∈ S. Let P be a proper (R, S)submodule of M . By Khumprapussorn et al. [7], a proper (R, S)-submodule P of M is called jointly prime if for each left ideal I of R, right ideal J of S, and (R, S)-submodule N of M with IN J ⊆ P implies either IM J ⊆ P or N ⊆ P . If R and S are commutative rings, we have a proper (R, S)-submodule P of M is called jointly prime if for each ideal I of R, ideal J of S, and (R, S)-submodule N of M with IN J ⊆ P implies either IM J ⊆ P or N ⊆ P . The concept of jointly prime (R, S)-submodules when R and S are arbitrary rings have been studied by Khumprapussorn et al. [7] and continued by Yuwaningsih and Wijayanti [8].
On module theory, a proper submodule N of an R-module M is called prime if for each element a of R and element m ∈ M with am ∈ N implies m ∈ N or aM ⊆ N . Prime submodules have been introduced and studied by Dauns [6]. As time went by, the researchers began to generalize the definition of prime submodules to weakly prime submodules. A proper submodule N of M is called weakly prime if for each submodule P of M and elements a, b of R satisfy abP ⊆ N , implies either aP ⊆ N or bP ⊆ N . Weakly prime submodules have been introduced by Behboodi and Koohy [4]. Moreover, the studied about weakly prime submodules have been continued by Azizi [2], Behboodi [5], and Azizi [3].
In this paper, we present some properties of left weakly jointly prime (R, S)submodules as the generalization of jointly prime (R, S)-submodules. A proper (R, S)-submodule P of M is called left weakly jointly prime if for each (R, S)submodule N of M and elements a, b of R such that abN S ⊆ P implies either aN S ⊆ P or bN S ⊆ P . Moreover, we present some properties of left weakly jointly prime (R, S)-submodules. Some of these properties are as follows: a proper (R, S)-submodules is left weakly jointly prime if and only if the annihilator of this quotient (R, S)-module over ring R is a prime ideal of R; when we give a left weakly jointly prime (R, S)-submodule, the set of that annihilator over ring R form a chain of prime ideals; if a left weakly jointly prime (R, S)-submodule is irreducible then it is a jointly prime; any left weakly jointly prime (R, S)-submodule P of M contains a minimal left weakly jointly prime (R, S)-submodule; and every left weakly jointly prime (R, S)-submodule is equal to jointly prime (R, S)-submodule in left multiplication (R, S)-module.

LEFT WEAKLY JOINTLY PRIME (R, S)-SUBMODULES
Before we present the definition of left weakly jointly prime (R, S)-submodules, we describe first the jointly prime (R, S)-submodule. As we have already stated earlier, when R and S are commutative rings, a proper (R, S)-submodule P of M is called jointly prime if for each ideal I of R, ideal J of S, and (R, S)-submodule N of M with IN J ⊆ P implies either IM J ⊆ P or N ⊆ P . When R and S are arbitrary rings, Khumprapussorn et al. [7] have given some characteristic of jointly prime (R, S)-submodules. In this paper, we modify those characteristics when R and S are commutative rings as follows. (1) P is jointly prime.
(2) For all ideal I of R, m ∈ M , and ideal J of S, ImJ ⊆ P implies IM J ⊆ P or m ∈ P . (3) For all a ∈ R, m ∈ M , and b ∈ S, amb ∈ P implies aM b ⊆ P or m ∈ P . (4) For all a ∈ R and m ∈ M , amS ⊆ P implies aM S ⊆ P or m ∈ P .
A proper submodule P of R-module M is called weakly prime if for each submodule K of M and elements a, b of R such that abK ⊆ P implies either aK ⊆ P or bK ⊆ P . This definition has studied by Azizi [2]. Based on that definition, we present the definition of left weakly jointly prime (R, S)-submodules as follows.
Note that right weakly jointly prime (R, S)-submodules can be defined and studied analogously. Now, if we have a condition a ∈ RaS for all a ∈ M , then we give the definition of left weakly jointly prime (R, S)-submodules as follows.
We can easily show that the two definitions of left weakly jointly prime (R, S)submodule above are equivalent. Moreover, we give some example of left weakly jointly prime (R, S)-submodules as follows.
Example 2.4. Let Z be an (2Z, 2Z)-module and 2Z be an (2Z, 2Z)-submodule of Z. We can show that 2Z is a left weakly jointly prime (2Z, 2Z)-submodule. Let any a, b ∈ 2Z with a = 2k and b = 2l and let any (2Z, 2Z)-submodule N = xZ of Z, for some k, l, x ∈ Z. Clearly that abN Example 2.5. Let R and S are commutative rings with Easily we can check that an (R, S)-submodule X of M with is a left weakly jointly prime (R, S)-submodule of M .

SOME PROPERTIES OF LEFT WEAKLY JOINTLY PRIME (R, S)-SUBMODULES
In this section, we present some properties of left weakly jointly prime (R, S)submodules. However, before we recall the definition of annihilator of the quotient (R, S)-modules over a ring R as follows.     Proof.
(2 ⇒ 1). Let r 1 , r 2 ∈ R and a ∈ M where r 1 r 2 aS ⊆ N . Since S 2 = S, then r 1 r 2 aSS ⊆ N . If r 1 aS N , we will show that r 2 aS ⊆ N . From r 1 r 2 aSS ⊆ N , we obtain r 1 ∈ (N : R r 2 aS) \ (N : R a). Consequently, (N : R r 2 aS) = (N : R a). Put x = r 2 aS and y = a, then by our assumption we get N = (N +Rr 2 aS 2 )∩(N +RaS). Since r 2 aS ⊆ Rr 2 aS 2 ⊆ N + Rr 2 aS and r 2 aS ⊆ RaS ⊆ N + RaS, we obtain r 2 aS ⊆ N .
Before we present the next properties, we recall the definition of an irreducible (R, S)-submodule as follow.     The following lemma gives us a property about the necessary and sufficient conditions of left weakly jointly prime (R, S)-submodules.  Proof. Let any weakly jointly prime (R, S)-submodule P of M and let J be the set of all weakly jointly prime (R, S)-submodules of M that contained in P . Clearly, J = ∅ since P ∈ J. By using Zorn's Lemma, we will show that J contains a minimal element. Equivalently, we show that every nonempty chain in J has a lower bound in J. Let any nonempty chain G ⊆ J. We can construct the set Q = K∈G K.
Then, clearly Q is an (R, S)-submodule of M and Q ⊆ P . We claim that Q is a weakly jointly prime (R, S)-submodule of M . Let any ideal I, J of R and an (R, S)-submodule N of M such that IJN S ⊆ Q but JN S Q. We will show that IN S ⊆ Q. Let any element n ∈ JN S \ Q. Then, there exist K ∈ G such that n ∈ K . Since K is a left weakly jointly prime (R, S)-submodule of M , then from IJN S ⊆ Q ⊆ K implies IN S ⊆ K . Moreover, let any L ∈ G. Since G is a chain of J, then K ⊆ L or L ⊆ K . If K ⊆ L, then we obtain IN S ⊆ K ⊆ L. If L ⊆ K , then we get n ∈ L. Since L is a left weakly jointly prime (R, S)-submodule of M , then from IJN S ⊆ Q ⊆ L implies IN S ⊆ L. Thus, we obtain IN S ⊆ L for any L ∈ G and so IN S ⊆ Q. Hence, proved that Q is a left weakly jointly prime (R, S)-submodule of M . Since Q ⊆ P , then Q ∈ J and Q is a lower bound of G. Thus, it's proved that every nonempty chain of J has a lower bound in J. Based on Zorn's Lemma, there exist a left weakly jointly prime (R, S)-submodule P * ∈ J that minimal among the left weakly jointly prime (R, S)-submodules in J. Thus, it's proved that any left weakly jointly prime (R, S)-submodules P contain minimal left weakly jointly prime (R, S)-submodule P * of M .
From Khumprapussorn et al. [7], we know that an (R, S)-module M is called left multiplication (R, S)-module provided that for each (R, S)-submodule N of M there exists an ideal I of R such that N = IM S. We have the characterization of jointly prime (R, S)-submodule of left multiplication (R, S)-modules as follow. In the following proposition, we present that every left weakly jointly prime (R, S)-submodules is equal to jointly prime (R, S)-submodules in left multiplication (R, S)-modules.

CONCLUDING REMARKS
Further work on the properties of left weakly jointly prime (R, S)-submodules can be carried out. For example, the investigation of properties of left weakly jointly prime radicals of an (R, S)-module.