International Journal of Algebra, Vol. 5, 2011, no. 4, 189 - 198

Graded Almost Prime Submodules

Malik Bataineh

Mathematics Department Jordan University of Science and Technology Irbid 22110, Jordan [email protected]

Ala’ Lutfi Khazaa’leh

Mathematics Department Jordan University of Science and Technology Irbid 22110, Jordan

Ameer Jaber

Department of Mathematics The Hashemite University Zarqa 13115, Jordan

Abstract

Let G be an abelian group with identity e, R be a G−graded com- mutative ring and M a graded R−module. A proper graded submodule N of M is a graded almost prime if for r ∈ h(R) and m ∈ h(M), rm ∈ N − (N : M)N implies r ∈ (N : M)orm ∈ N. A proper graded submodule is a graded weakly prime submodule if for r ∈ h(R) and m ∈ h(M), 0 = rm ∈ N implies r ∈ (N : M)orm ∈ N. Ev- ery graded prime submodule is graded weakly prime as well as graded almost prime. However, since {0} is always a graded weakly prime sub- module and hence a graded almost prime submodule, a graded almost prime submodule need not be a graded prime. Throughout this work, we define graded almost prime submodules as a new generalization of graded prime and graded weakly prime sub- modules of a graded unital modules over a graded commutative ring. We study some properties of graded almost prime submodules and give some of their characterizations.

Mathematics Subject Classification: 13A02, 16W50 190 M. Bataineh et al

Keywords: Graded Rings, Graded Modules, Graded almost prime Sub- modules, Graded weakly prime Submodules

1 Introduction

Several authors have extended the notion of prime ideals to modules [3], [5]. Almost prime ideals were introduced by S M. Bhatwadekar and P. K. Sharma [9] and later studied by D. D. Anderson, and M. Bataineh [7]. Weakly prime ideals were introduced by A. G. Agargun, D. D. Anderson, and S. Valdes-Leon [1] and later studied by D. D. Anderson and E. Smith [4]. A proper ideal P of R considered as almost prime ideal if ab ∈ P −P 2 then a ∈ P or b ∈ P . Graded almost prime ideals in a graded commutative ring with non-zero identity have been introduced and studied by A. Jabeer, M. Bataineh and H. Khashan [2, 6]. Moreover, graded weakly prime submodules in a graded commutative ring with non-zero identity have been introduced and studied by S. E. Atani [11]. In this study, we investigated graded almost prime submodules of a graded module over a graded commutative ring. A proper graded submodule N of a unital graded module is considered a weakly prime if for a ∈ h(R) and m ∈ h(M), 0 = am ∈ N implies a ∈ (N : M)orm ∈ N . A proper graded submodule of a unital graded module is graded almost prime if for a ∈ h(R) and m ∈ h(M), am ∈ N − (N : M)N implies that a ∈ (N : M)orm ∈ N. Throughout this work we investigate various properties of graded almost prime submodules and their generalizations. Moreover, for a graded almost prime submodule N in a unital graded module M over a graded commutative ring R, we show that S−1N is a graded almost prime submodule for any multiplicatively closed subset S of R disjoint from (N : M). We also explore the relationship between graded almost prime and graded weakly prime submodules.

2 Graded Almost Prime Submodules

In this chapter, the authors define graded almost prime (g-prime) submodules, give some properties, characterizations and examples of graded almost prime (g-prime) submodules and show relationship between graded weakly prime (g-prime) submodules and graded almost prime (g-prime) submodules. Let us introduce some notation and terminology. Let G be an arbitrary abelian group with identity e.ByaG−graded commutative ring we mean a commutative ring R with non-zero identity together with a direct sum decom- position (as an additive group) R = ⊕ Rg with the property that RgRh ⊆ Rgh g∈G for all g, h ∈ G . Also, we write h(R)=∪g∈GRg. The summands Rg are called homogeneous components and elements of these summands are called homo- geneous elements Moreover, Re is a subring of R and 1R ∈ Re. Product of graded submodules 191

Let R be a G−graded ring and M an R−module. We say that M is a G−graded R−module if there exists a family of additive subgroups {Mg}g∈G of M such that M = ⊕g∈GMg (as abelian groups), andRgMh ⊆ Mgh for all g, h ∈ G. Also, we write h(M)=∪g∈GMg.IfM = ⊕g∈GMg is a graded R-module, then Mg is an Re−module for all g ∈ G. Let N be a graded submodule of M, and let g ∈ G.There are some conditions equivalent to Ng being as weakly g-prime. For example, Ng is weakly g-prime if and only if for P is an ideal of Re and K is a submodule of Mg,0= PK ⊆ Ng implies P ⊆ (Ng : Mg)orK ⊆ Ng. The next theorem has been proved in [11, Theorem 2.6]. Theorem 2.1 [11] For a proper graded submodule N of M and g ∈ G. Then the following are equivalent. (1) If whenever 0 = PK ⊆ Ng with P an ideal of Re and K a submodule of Mg, then P ⊆ (Ng : Mg) or K ⊆ Ng. (2) Ng is a weakly g-prime submodule of Mg. (3) For m ∈ Mg − Ng, then (Ng : m) ≡ (Ng : Mg) ∪ (0 : m). (4) For m ∈ Mg − Ng, then (Ng : m) ≡ (Ng : Mg) or (Ng : m) ≡ (0 : m).

Next we define an almost g-prime Re-submodule Ng of Mg and the graded almost prime submodule N of M. Definition 2.2 Let N be a graded submodule of M and g ∈ G. (1) Ng is an almost g-prime submodule of the Re-module Mg ,ifNg = Mg ; and whenever a ∈ Re and m ∈ Mg with am ∈ Ng − (Ng : Mg)Ng then either a ∈ (Ng : Mg) or m ∈ Ng. (2) N is a graded almost prime submodule of M,ifN = M; and whenever a ∈ h(R),and m ∈ h(M) with am ∈ N − (N : M)N, then either a ∈ (N : M) or m ∈ N. Clearly a graded prime (resp. g-prime) submodule is a graded weakly prime (resp. weakly g-prime) submodule and a graded weakly prime (resp. weakly g- prime) submodule is a graded almost prime (resp. almost g-prime) submodule. However, since {0} is a graded weakly prime submodule and hence a graded almost prime submodule (by definition), a graded weakly prime submodule and a graded almost prime submodule need not be graded primes. Now we give a non trivial example of a graded almost prime submodule that is not a graded prime submodule.

Example 2.3 Let R ≡ Z[i]=Z + iZ, G = Z2 and M = Z54[i]. Then M is a graded R−module with M0 = Z54 and M1 = iZ54. Take N = 27 + i27 . Then (N : M)=27R and (N : M)N = N.So clearly Nis a graded almost prime submodule, but not a graded weakly prime submodule since 0 =9 i(3) = 27i ∈ N but neither 9i ∈ (N : M) nor 3 ∈ N. Hence Nis not graded prime. 192 M. Bataineh et al

Next we give an example of a graded almost prime submodule that is not almost prime.

Example 2.4 Let R ≡ Z[i]=Z + iZ, G = Z2 and M is R−module itself. N =2M is a graded submodule of M. Then (N : M)=2R and (N : M)N = 4M. Check that N is a graded almost prime submodule, but N as a submodule of M is not almost prime since (1 −i)(1 + i)=2∈ N −(N : M)N and neither (1 − i) ∈ (N : M) nor (1 + i) ∈ N.

We next give four characterizations of homogeneous components of the graded submodules. Compare this result with [11, Theorem 2.6].

Theorem 2.5 Let N be a graded submodule of M. Then the following are equivalent. (1) Ng is a graded almost prime submodule of Mg. (2) For m ∈ Mg − Ng, then (Ng : m) ≡ (Ng : Mg) ∪ ((Ng : Mg)Ng : m). (3) For m ∈ Mg − Ng, then (Ng : m) ≡ (Ng : Mg) or (Ng : m) ≡ ((Ng : Mg)Ng : m). (4) If whenever PK ⊆ Ng and PK (Ng : Mg)Ng with P an ideal of Re and K a submodule of Mg, then P ⊆ (Ng : Mg) or K ⊆ Ng.

Proof. (1 ⇒ 2) Let m ∈ Mg − Ng and a ∈ (Ng : m), then am ∈ Ng.If am∈ / (Ng : Mg)Ng, then a ∈ (Ng : Mg) since Ng is graded almost prime and m/∈ Ng.Ifam ∈ (Ng : Mg)Ng implies a ∈ ((Ng : Mg)Ng : m). The reverse inclusion if a ∈ (Ng : Mg), then aMg ⊆ Ng,soam ∈ Ng. Hence a ∈ (Ng : m). If a ∈ ((Ng : Mg)Ng : m), then am ∈ (Ng : Mg)Ng ⊆ Ng , a ∈ (Ng : m). Therefore (Ng : m) ≡ (Ng : Mg) ∪ ((Ng : Mg)Ng : m). (2 ⇒ 3) Clearly, from the graded ring theory on graded ideals. (3 ⇒ 4) Let P be an ideal of Re and K be a submodule of Mg such that PK ⊆ Ng, P (Ng : Mg) and K Ng. Want PK ⊆ (Ng : Mg)Ng. Let a ∈ P and m ∈ K. Case 1 assume that m/∈ Ng.Ifa/∈ (Ng : Mg). Since am ∈ Ng, we have (Ng : m) =( Ng : Mg). By assuming (Ng : m) ≡ ((Ng : Mg)Ng : m). So am ∈ (Ng : Mg)Ng.Ifa ∈ P ∩ (Ng : Mg). Let b ∈ P − (Ng : Mg). Then a + b ∈ P − (Ng : Mg). By previous case, we have bm ∈ (Ng : Mg)Ng and (a + b)m ∈ (Ng : Mg)Ng.Soam ∈ (Ng : Mg)Ng. Case 2 assume that m ∈ Ng. Let m ∈K − Ng. Then m + m ∈K − Ng. By Case 1 am ∈(Ng : Mg)Ng and a(m + m ) ∈ (Ng : Mg)Ng. Hence am ∈ (Ng : Mg)Ng for arbitrary a ∈ P and m ∈ K. Therefore PK ⊆ (Ng : Mg)Ng. (4 ⇒ 1) Let am ∈ Ng − (Ng : Mg)Ng with a ∈ Re and m ∈ Mg. Take P ≡ Rea and K ≡ Rem, then PK ⊆ Ng and PK (Ng : Mg)Ng.Soby assuming P ⊆ (Ng : Mg)orK ⊆ Ng. Hence a ∈ (Ng : Mg)orm ∈ Ng. 

Easy computations give us the following lemma. Product of graded submodules 193

Lemma 2.6 If N is a graded almost prime submodule of M, then Ng is an almost g-prime submodule of Mg for all g ∈ G. 

But the converse of the last theorem is not necessarily true. √ Example 2.7 Let√R ≡ Z[ 3], G ≡ Z2 and M√is R−module itself. Take N =3√ M =3Z +3 3Z, then (N :√M)=3Z[ 3] and so (N : M)N = 9Z[ 3]. Clearly N 0 ≡ 3Z and N1 ≡ 3 3Z are g-prime submodules of Z, and also√ almost√ g-prime submodules. But N is not√ graded almost prime√ . Since (2 3)( 3) = 6 ∈ N − (N : M)N with neither 2 3 ∈ (N : M) nor 3 ∈ N.

The proof of the next result is the same as the proof of Theorem 2.5. So we state the next result without proof.

Theorem 2.8 Let N be a graded submodule of M. Then the following are equivalent. (1) N is a graded almost prime submodule of M. (2) For m ∈ h(M) − h(N), then (N : m) ≡ (N : M) ∪ ((N : M)N : m). (3) For m ∈ h(M) − h(N), then (N : m) ≡ (N : M) or (N : m) ≡ ((N : M)N : m). (4) If whenever PK ⊆ N and PK (N : M)N with P an ideal of R and K a submodule of M, then P ⊆ (N : M) or K ⊆ N. 

Next we have another four characterizations of graded almost prime submod- ules.

Theorem 2.9 Let N be a graded submodule of M . Then the following are equivalent. (1) N is a graded almost prime submodule of M. (2) For A ⊆ M and A N, then (N : A) ≡ (N : M) ∪ ((N : M)N : A) where A is a submodule of M. (3) For A ⊆ M and A N, then (N : A) ≡ (N : M) or (N : A) ≡ ((N : M)N : A) where A is a submodule of M. (4) If whenever PK ⊆ N and PK (N : M)N with P an ideal of R and K a submodule of M, then P ⊆ (N : M) or K ⊆ N.

Proof. (1 ⇒ 2) Suppose that A ⊆ M and A N. Let a ∈ (N : A), then aA ⊆ N.Soa A ⊆ N.Ifa A (N : M)N, then a ⊆(N : M) since N is graded almost prime and A N. And so a ∈ (N : M). If a A ⊆ (N : M)N implies a ∈ ((N : M)N : A). The reverse inclusion if a ∈ (N : M), then aM ⊆ N,soam ∈ N ∀ m ∈ M. Hence a ∈ (N : A). If a ∈ ((N : M)N : A), then aA ⊆ (N : M)N ⊆ N, a ∈ (N : A). Therefore (N : A) ≡ (N : M)∪((N : M)N : A). (2 ⇒ 3) Clearly, from the graded ring theory on graded ideals. 194 M. Bataineh et al

(3 ⇒ 4) Let P be an ideal of R and K be a submodule of M such that PK ⊆ N, P (N : M) and K N. Want PK ⊆ (N : M)N. Let a ∈ h(P ) and m ∈ h(K). Case 1 assume that m/∈ N.Ifa/∈ (N : M). Since am ∈ N, we have (N : m) =( N : M). By assuming (N : m) ≡ ((N : M)N : m). So am ∈ (N : M)N.Ifa ∈ P ∩ (N : M). Let b ∈ P − (N : M). Then a + b ∈ P − (N : M). By previous case, we have bm ∈ (N : M)N and (a + b)m ∈ (N : M)N.Soam ∈ (N : M)N. Case 2 assume that m ∈ N. Let m ∈K − N. Then m + m ∈K − N.By Case 1 am ∈(N : M)N and a(m + m ) ∈ (N : M)N. Hence am ∈ (N : M)N for arbitrary a ∈ h(P ) and m ∈ h(K). And so h(P )h(K) ⊆ (N : M)N. Therefore PK ⊆ (N : M)N. (4 ⇒ 1) Let am ∈ N − (N : M)N with a ∈ h(R) and m ∈ h(M). Take P ≡a and K ≡ Rm, then PK ⊆ N and PK (N : M)N. So by assuming P ⊆ (N : M)orK ⊆ N. Hence a ∈ (N : M)orm ∈ N. 

We present some properties of graded almost prime submodules and almost g-prime submodules in graded modules.

Remark 2.10 Let N and K be two submodules of M such that K ⊆ N, then (N : M)=(NK : MK).

Theorem 2.11 Assume that N and K are graded submodules of M such that K ⊆ N with N = M. Then (1) If N is a graded almost prime submodule of M, then NK is a graded almost prime submodule of MK. (2) If K and NK are graded almost prime submodules, then N is a graded almost prime submodule of M.

Proof. (1) Let a(m + K) ≡ am + K ∈ NK − (NK : MK)NK where a ∈ h(R) and m ∈ h(M). Then am + K ∈ NK − (N : M)NK and so am ∈ N.Ifam ∈ (N : M)N, then am + K ∈ (N : M)NK which is a contradiction. If am∈ / (N : M)N, N is a graded almost prime submodule gives either a ∈ (N : M)orm ∈ N; hence either m + K ∈ NK or a ∈ (NK : MK). (2) Let am ∈ N−(N : M)N where a ∈ h(R),and m ∈ h(M), so a(m+K) ∈ NK.Ifam ∈ K, since am∈ / (N : M)N and K ⊆ N then (K : M) ⊆ (N : M)soam∈ / (K : M)K. Hence am ∈ K − (K : M)K. Since K is graded almost prime gives either a ∈ (K : M) ⊆ (N : M)orm ∈ K ⊆ N.Ifam∈ / K, we get am∈ / (N : M)NK. Then a(m + K) ∈ NK − (N : M)NK. Since NK is a graded almost prime submodule we get either m ∈ N or a ∈ (NK : MK)=(N : M). 

If K ⊆ N ⊂ M are submodules of M, then N is a graded prime submodule Product of graded submodules 195 of M if and only if NK is graded prime of MK. While in the graded almost prime submodules case, N is a graded almost prime submodule of M if and only if NK is a graded almost prime submodule of MK, for any submodule K ⊆ N, the converse part is true only when K is a graded almost prime submodule. For example, for any non graded almost prime submodule N of M, we have 0 = NN is a graded weakly prime (and so graded almost prime) submodule of MN. Compare the next theorem with [11, Theorem 2.17].

Theorem 2.12 Assume that N and K are graded almost prime submodules of M such that N + K = M. Then N + K is a graded almost prime submodule of M . ∼ Proof. Since (N + K)/K = K/(N ∩ K), we get (N + K)/K is a graded almost prime submodule by Theorem 2.11(1). Now the assertion follows from Theorem 2.11(2). 

In the following theorem, we show the relationship between graded almost prime and graded weakly prime submodules.

Theorem 2.13 Let N be a graded submodule of M and g ∈ G. Then Ng is an almost g-prime submodule in Mg if and only if Ng(Ng : Mg)Ng is a weakly g-prime submodule in Mg(Ng : Mg)Ng.

proof Let r ∈ Re and m ∈ Mg such that 0 = r(m +(Ng : Mg)Ng) ∈ Ng(Ng : Mg)Ng, then rm ∈ Ng − (Ng : Mg)Ng.Som ∈ Ng or r ∈ (Ng : Mg) since Ng is almost g-prime in Mg. Then m+(Ng : Mg)Ng ∈ Ng(Ng : Mg)Ng or r ∈ (Ng : Mg) ≡ ( Ng(Ng : Mg)Ng: Mg(Ng : Mg)Ng). Conversely, let r ∈ Re and m ∈ Mg such that rm ∈ Ng − (Ng : Mg)Ng. Then 0 = r(m +(Ng : Mg)Ng) ∈ Ng(Ng : Mg)Ng.Som ∈ Ng or r ∈ (Ng(Ng : Mg)Ng: Mg(Ng : Mg)Ng) ≡ (Ng : Mg). 

Because the proof of the next theorem is the same as the proof of Theorem 2.13, we state the following result without proof.

Theorem 2.14 Let N be a graded submodule of M. Then N is a graded almost prime submodule of M if and only if N(N : M)N is a graded weakly prime submodule of M(N : M)N. 

Definition 2.15 A subset S ⊆ h(R) is said to be multiplicatively closed subset if for any a,b ∈ S implies ab ∈ S.

Remark 2.16 Let N be a graded submodule of a graded R-module M and S a multiplicatively closed subset of R. Then (N : M) ⊆ (S−1N : S−1M). 196 M. Bataineh et al

Proposition 2.17 Let N be a graded almost prime submodule of M. Then if S ⊆ h(R) is multiplicatively closed subset of R with (N : M) ∩ S = φ, then S−1N is a graded almost prime submodule of S−1M. ∈ m ∈ −1 m ∈ −1 − −1 −1 proof Let r h(R) and s h(S M) with r s S N (S N : S M). rm ∈ −1 − −1 rm ∈ −1 ∈ Then s S N S ((N : M)N). As s S N, there is x S and ∈ rm n ∈ ∈ n h(N) such that s = x and there is a S such that arxm = asn N. Moreover, for any b ∈ S, brm∈ / (N : M)N.Soaxrm ∈ N − (N : M)N. Since N is graded almost prime, then either ax ∈ (N : M)orrm ∈ N. Since ax∈ / (N : M), then rm ∈ N and so rm ∈ N − (N : M)N . Hence ∈ ⊆ −1 −1 ∈ m ∈ −1  r (N : M) (S N : S M)orm N (implies s S N ).

Let N be a graded submodule of a graded R−module M. Then the quotient MN is a graded R(N : M)−module with the action of ring elements is (r + P )(m + N)=rm + N where r ∈ h(R), m ∈ h(M) and P =(N : M). Theorem 2.18 Let N be a graded almost prime submodule of M with (N : M)=P . Then there is a one-to-one correspondence between graded almost prime submodules of the RP −module MN and the graded almost prime submodules of the graded R−module M containing N. proof Let K be a graded submodule of M containing N . Suppose that K is a graded almost prime. We want KN is a graded almost prime submodule of MN.NowKN is a proper submodule of MN since K is proper of M. Let (a + P )(m + N) ∈ KN− (KN : MN)(KN ), then am + N ∈ KN− ((K : M)K)N) .am + N/∈ ((K : M)K)N ), so am∈ / (K : M)K implies that am ∈ K − (K : M)K. Since K is a graded almost prime submodule we obtain a ∈ (K : M)orm ∈ K. Therefore, a + P ∈ (K : M)P ≡ (KN : MN)P or m + N ∈ KN. Conversely, suppose that KN is a graded almost prime submodule. Want K is a graded almost prime submodule. Let am ∈ K − (K : M)K such that a ∈ h(R) and m ∈ h(M). If a ∈ N, then am ∈ N − (N : M)N since N ⊆ K.AsN is a graded almost prime submodule gives m ∈ N ⊆ K or a ∈ (N : M) ⊆ (K : M). If a/∈ N, then (a + P )(m + N) ∈ KN− (KN : MN)(KN). Since KN is a graded almost prime submodule then a + P ∈ (KN : MN)P or m + N ∈ KN,soa ∈ (K : M)or m ∈ K.  Definition 2.19 Let M be a graded R-module, we say that a non-zero homogeneous element x ∈ h(M) is almost zero divisor on M if there is 0 = r ∈ h(R) such that rx =0. The next proposition has been proved for graded almost prime ideals by H. Khashan, A. Jaber and M. Bataineh in [6]. Now we extend it for graded almost prime submodules. Product of graded submodules 197

Proposition 2.20 Let N be a graded almost prime submodule of a graded R-module M. Then (1) If x ∈ h(M) is an almost zero divisor on (R/(N : M))−module MN, then either x ∈ N or (N : M)x ⊆ (N : M)N. (2) If K is a graded submodule consists of almost zero divisor on MN and K ⊆ N, then (N : M)K =(N : M)N.

proof (1) Since x is an almost zero divisor on MN, there is r ∈ h(R) − h(N : M) such that rx ∈ N. Suppose that x/∈ N. Since N is a graded almost prime submodule, then rx ∈ (N : M)N. Let a ∈ (N : M), then am ∈ N for all m ∈ M. We have ax ∈ N, then (a + r)x ∈ N and (a + r) ∈/ (N : M). Since N is a graded almost prime submodule, then (a + r)x ∈ (N : M)N. Hence ax ∈ (N : M)N for any a ∈ (N : M). Therefore, (N : M)x ⊆ (N : M)N. (2) Let r ∈ (N : M) and x ∈ K. Want rx ∈ (N : M)N. Since x is an almost zero divisor on MN , then by(1) x ∈ N or (N : M)x ⊆ (N : M)N. If x ∈ N, then rx ∈ (N : M)N.If(N : M)x ⊆ (N : M)N, then rx ∈ (N : M)x ⊆ (N : M)N. 

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Received: September, 2010