Pure and Applied Mathematics Journal 2015; 4(2): 47-51 Published online February 11, 2015 (http://www.sciencepublishinggroup.com/j/pamj) doi: 10.11648/j.pamj.20150402.13 ISSN: 2326-9790 (Print); ISSN: 2326-9812 (Online) Graded Essential Extensions and Graded Injective Modules Salah El Din S. Hussein, Essam El Seidy, H. S. Diab* Department of Mathematics, Faculty of Science, Ain Shams University, Abasaya-Cairo, Egypt Email address: [email protected] (S. E. D. S. Hussein), [email protected] (E. E. Seidy), [email protected] (H. S. Diab) To cite this article: Salah El Din S. Hussein, Essam El Seidy, H. S. Diab. Graded Essential Extensions and Graded Injective Modules. Pure and Applied Mathematics Journal. Vol. 4, No. 2, 2015, pp. 47-51. doi: 10.11648/j.pamj.20150402.13 Abstract: In this paper we establish the relation between graded essential extensions of graded modules and injectivity of such modules. We relate the graded injective hull Egr (M) of a graded module M with the graded essential extensions of M. We round off by establishing necessary and sufficient conditions for indecomposability of graded injective modules in terms of their graded injective hulls. Keywords: Graded Injective Module, Graded Essential Extension, Graded Injective Hull = ⊕ identity e and R R σ is a G-graded ring. The elements 1. Introduction σ∈G U Rσ of are called homogeneous elements of R. All The study of projective and injective modules constitutes σ∈G the backbone of the modern homological algebra. Injective R-modules are left R-modules and by R-mod we denote the modules are closely related to essential extensions. Actually, category of left R-modules. An R-module M is said to be if R is an associative ring with 1 then an R-module M is = ⊕ G-graded if M M σ , for additive subgroups M of injective if and only if it has no proper essential extension. σ∈G σ Moreover, the idea of essential extensions has been proved to M, satisfying Rσ M τ⊂ M στ for all σ, τ ∈G . be indispensable in the formation of the injective hull of a For any G-graded R-modules M and N, an R-linear map module which is crucial for the theory of rings of quotient. f: M→ N is said to be a graded morphism of degree For over forty years, the theory of injective modules has been τ, τ ∈G , if f( Mσ ) ⊂ N στ for all σ ∈G . By R-gr we widely investigated and various aspects of this theory has denote the category of G-graded R-modules and the been completely settled, cf. [1],[2],[3],[4]. morphisms are taken to be the graded morphisms of degree e. This paper is devoted to the study of the interplay between If f: M→ N is a graded morphism of degree e we say graded essential extensions and graded injective modules. In section 2 we recall some basic notions, definitions, and that f is a gr-homomorphism. A submodule N of a G-graded preliminary results. In section 3 we investigate the relation R-module M is said to be a graded submodule of M if N= ⊕ ( NI M σ ) ∈ between the concept of graded essential extensions and σ∈G , or equivalently, if for any x N the injectivity of graded modules. Actually, after deriving the homogenous components of x are again in N. By basic properties of graded essential extensions in Proposition N≤gr M we denote a graded submodule N of the graded 3.1, we provide an important characterization of graded ∈ − K≤ M injective modules in terms of graded essential extensions. module M. If M R gr and gr then by Zorn's Graded versions of the well known results of Echmann, lemma there exist a graded submodule C of M which is Schopf, Baer, Bass, and Papp are proved in Theorem 3.4 and maximal with respect to KI C = 0 . The submodule C is Theorem 3.8. We round off by establishing equivalent called a graded complement of K in M. We say that conditions for indecomposability of graded injective modules M∈ R − gr is gr-injective if for any gr-monomorphism (Theorem 3.11). α : N→ M and any gr-homomorphism β : N→ L , where N, L∈ R − gr , there exists a gr-homomorphism γ : M→ L 2. Preliminaries such that β= γα . As in the ungraded one can easily observe Throughout this paper G is a multiplicative group with that a graded R-module M is gr-injective if and only if, 48 Salah El Din S. Hussein et al. : Graded Essential Extensions and Graded Injective Modules K≤ N ⊕ ≤ ess . whenever gr in R-gr, any gr-homomorphism (i) K Cgr M f: K→ M can be extended to a gr-homomorphism ess . (ii) (KCC⊕ )/ ≤ gr MC / . f' : N→ M . K≤ess . N N≤ess . M For details on graded rings and modules we refer to [5],[6], Proof. 1. Suppose that gr and gr . If while for details on extended injectivities of modules we refer 0≠x ∈ h ( M ) then there exists a∈ h( R ) such that to [7],[8] ess . 0≠ax ∈ h ( N ) . But N≤ M implies that 0≠bax ∈ h ( M ) , Lemma 2.1. (Baer criterion) A graded R-module M is gr ess . gr-injective if and only if whenever I is a graded left ideal of R, for some b∈ h( R ) . This entails that K≤gr M . every gr-homomorphism f: I→ M extends to a ess . Conversely, if K≤gr M and 0≠m ∈ h ( N ) then there gr-homomorphism f' : R→ M . exist c∈ h( R ) for which 0≠cm ∈ hK () ⊆ hN () . Thus Proof. Similar to the proof of Baer's theorem in the ess . ungraded case, cf. [7] and [8]. N≤gr M , since hN()⊂ hM () , obvious. Remark 2.2. The extension of f to f ' in Lemma 2.1 KKI≤ NN I 2. We first observe that 1gr 1 . Indeed, means exactly to find an element m= f'(1) ∈ M such that for every x∈ Ifx, ( ) = fx '( ) = xf . '(1) and then KKI= ⊕ [( KNM IIIIIσ ) ( KNM σ )] 1σ∈G 1 1 = f( x ) xm . , thus to extend f to f ' is equivalent to the = ⊕ I I I I (KK1 ) [( NN 1 ) M σ ] condition that f is just a right multiplication by an element σ∈G ∈ f= ⋅ m = ⊕ (KKI )( I NN I ) σ m M (in such case we write ). Moreover, since σ∈G 1 1 1∈ Re and f ' is graded of degree e then 0≠x ∈ hN (I N ) x∈ h( N ) f'(1)= mM ∈e ⊂ hM ( ) . Now, if 1 then and x∈ h( N ) ∈ 1 . By assumption, there exist a h( R ) such that ≠ ∈ 0≠ax ∈ h ( N ) x∈ h( N ) 3. Graded Essential Extensions and 0ax h ( K ) . Since 1 (Note that 1 ax∈ h( N ) N Graded Injectivity then 1 as 1 is an R-module) then there exists ∈ 0≠bax ∈ h ( K ) ≠ ∈ If N is a graded submodule of a graded R-module M then N b h( R ) such that 1 . Obviously, 0ba h ( R ) ≠ ∈ 0≠bax ∈ hK (I K ) is called a gr-essential submodule of M, or M is called a and 0bax h ( K ) . Thus, 1 , or equivalently ≠ gr-essential extension of N if NI X 0 for every non-zero KKI≤ess . NN I 1gr 1 . graded submodule X of M. This state of affairs is denoted by −1 ess . ess . 3. We first claim that α (K ) ≤gr M . Indeed, if N≤gr M . Obviously N≤gr M if and only if for each n −1 ≠ ∈ xx=∑ σ ∈α( K ), σ ∈ G for each i, then 0 x in h(M) there exists an a h( R ) such that i=1 i i n n 0≠ax ∈ h ( N ) . ≤α = α ∈ α()x=∑ ( x ) = ∑ (()) α x K Nx, (σ ) () xσ K σ σ . Since gr i i In the next proposition we state and prove the graded i=1i i = 1 i −1 version of the well known basic properties of essential for each i. Thus xσ ∈α ( K ) for each i and we have done. extensions. i Proposition 3.1. Let M be a graded R-module Let 0≠x ∈ h ( M ) . If α(x )= 0 then aα( x )= 0 ∈ K for ess . 1) If K≤gr N ≤ gr M then K≤gr M if and only if each a∈ h( R ) . It follows that there exists b∈ h( R ) such ess . ess . −1 α ≠ K≤gr N and N≤gr M . that 0≠bx ∈ h (α ()) K . But if (x ) 0 then it is obvious, ess. ess . by hypothesis that there exist c∈ h( R ) such that 2) If K≤≤gr N gr MK& ≤ gr N ≤ gr M then 1 1 − ≠ ∈ α 1 . This proves our claim. KKI≤ess . N ≤ NN I . 0cx h ( ( K )) 1gr gr 1 ⊕K ≤gr ⊕ M α ∈ ≤ess . ≤ 4. , 5. It is well known that iI∈i iI ∈ i and 3) If HomR− gr ( M , N ) and Kgr N gr N , then k⊕ CC/ ≤ gr MC / , cf.[5]. According to proposition 2.3.5 −1ess . α (K ) ≤gr M . of [6], the proof of 4. and 5. is similar to the proof of lemma 1.1 and lemma 1.7 of [3]. 4) Let M⊕ M , where M≤gr M for each i, and let i∈ I i i Lemma 3.2. Every graded R-module M has a maximal ess . K≤gr M for each i. Then ⊕K ≤ gr M if and only graded essential extension.