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.