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. i i i∈ I i Proof. Let I be a gr-injective module containing M. ≤ess . Λ = if Kigr M i for each i. Let {Ni } i∈ I be a family of graded essential 5) Let K≤ M . If C is a graded complement of K in M ≤ess . ≤ ∈ gr extensions of M such that Mgr Ni I , for every i I . then Suppose that Λ is linearly ordered by inclusion. Let Pure and Applied Mathematics Journal 2015; 4(2): 47-51 49

Γ = {N } ≤ INI =0, M ≤ I = j j∈ J be a totally ordered subset of Λ . For every N I . Since 1gr 1 then NI M 0 , we ≠ ∈ U ∈ = M≤ess I I= I 0x h ( N j ) x h( N j ) j∈ J must have N 0 (since gr ), so 1. j∈ J we have 0 for some 0 . 1 ≤ess . 2 ⇒ 3. We claim that any graded module M has a Since Mgr N j then there exists a∈ h( R ) such that R 0 graded maximal essential extension: M≤ess . U N U N Fix a gr-injective module I such that M⊂ I , and consider 0 ≠ax ∈ M , thus gr ∈ j . Hence ∈ j is an j J j J any family of gr-essential extensions of M in I that are linearly upper bound for Γ . Thus, by Zorn’s lemma, Λ has a ordered by inclusion, Let P be a chain of this family, it is clear maximal element E. If E ' is a graded R-module such that that the union is an upper bound of this chain and the union is ess . M≤gr E ' . If E⊆/ E ' then, by the injectivity of I the also gr-essential over M. By Zorn's Lemma, it follows that we → can find a submodule E maximal with respect to the property inclusion f: E I can be extended to a gr-homomorphism ess that M≤ E ≤ I . If I is a gr-minimal injective over M, g: E ' → I . Obviously, MI Kerg = 0 , and since gr gr then there exists a maximal essential extension submodule E ≤ess . = Mgr E ' then Kerg 0 . Thus g is a monomorphism. It over M such that M⊂ E ⊂ I . Using (1) ⇒ (2), we know follows that Eó E'≅ gE ( ') ≤ I which contradicts the that E is gr-injective, and by minimality of I, E = I, thus I is maximality of E in Λ . Therefore E is a maximal graded gr-maximal essential over M. ≤ essential extension of M. Definition 3.5. If the modules Mgr I satisfy one (and Theorem 3.3. Suppose that G is finite. A graded R-module hence all) of the equivalent properties of Theorem 3.4, we say M is gr-injective if and only if it has no proper gr-essential that I is a gr-injective hull (or gr-injective envelope) of M. extension. As we shall prove in Corollary 3.6 below; the gr-injective Proof. Let M be gr-injective. Suppose that there exists hull of M is unique up to isomorphism. The gr-injective hull of ∈ − ess . E R gr such that M

gr-injective. η → i:/AA i IAA (/) i Q:= ⊕ Q Proof. Let i∈ I i be an internal or external direct sum And let of gr-injective left R-modules Q . By Baer's Criterion, for i ∞ every left gr-ideal U≤ R and every gr-homomorphism α → ⊕ :A IAA (/)i ρ :U→ Q there exists a gr-homomorphism τ : R→ Q with i=1 ρ= τ → i where i: U R is the inclusion mapping. Since be defined by R is gr-noetherian, U is finitely generated: R na α():a=∑ ( aAaA + ) ∈ n i=1 i u= ∑ Ru i=1 i η = ≥ Then follows that i 0 for i n and consequently ρ = ρ The images (u ), i 1,2,3,..., n of the u under = ≥ i i A A i for i n have components different from zero for only finitely many of Theorem 3.9. Let R be a graded noetherian ring, then any i∈ I I the Qi , say for the Qi with 0 where 0 is a finite graded injective R- module is a direct sum of indecomposable graded injective submodules. io : ⊕ Q → ⊕ Q subset of I. Let iI∈o i iI ∈ i be the inclusion Proof. First we prove that any gr-injective module contains ρ gr-injective indecomposable submodule. Let mapping and let o be the gr-homomorphism, induced by I= E( Rx ), x ∈ hI ( ) . ρ ⊕ Q gr the restriction of the domain of to i∈ I o i then we have Since R is gr-noetherian then it is easily checked that ⊕ ρ= i ρ I Q Egr ( Rx ) contains an indecomposable gr-injective o o . Since o is finite, i∈ I o i is gr-injective and there submodule. Now, for any gr-injective module M , consider exists a gr-homomorphism τo so that the following diagram R is commutative: all the families of all gr -injective indecomposable submodules of M whose sum is direct. By Zorn's lemma there ∈ exists a family {Mi : i I } which is maximal, then = ⊕ ⊕ ≤ ⊕ M N(i M i ) for some Ngr M . Since iM i is gr-injective by Theorem 3.8 and E is a direct summand of gr-injective as well, then by the first part of the proof E must contian an indecomposable gr-injective submodule which ∈ contradicts the maximality of {Mi : i I } , thus E must be zero. This proves our claim. Corollary 3.10. Let N be a finitely generated module over a graded noetherian ring R. Then Egr ( N ) is a finite direct sum of indecomposable gr-injective submodules. E( N ) = ⊕ M ' Proof. By Theorem 3.9, gr ii where the Mi s are indecomposable gr-injective. Since N is finitely generated, ⊆ ⊕ ⊕ N Mi...... M i i,...... i we have 1 n for suitable 1 n .By Theorem 3.7, this finite direct sum is gr-injective. Therefore, = ⊕ ⊕ ENMgr ( )i ...... M i by Corollary 3.7, we must have 1 n . ρ= ρ = τ = τ Consequently we have io o iiio o if we put A nonzero graded module M is called graded uniform if any nonzero graded submodules of M intersect nontrivially τ:= i τ o o . (equivalently any nonzero graded submodule of M is Conversely, consider an arbitrary chain of left gr-ideals indecomposable). A graded ideal I≤gr R is called graded A≤ A ≤ A ≤ ... (R / I ) 1 2 3 meet-irreducible if the cyclic graded module R is uniform. And let Theorem 3.11. For any graded injective module M over a ∞ graded ring R , the following conditions are equivalent: A:= U A 1. M is gr- indecomposable. i=1 i 2. M ≠ 0 , and M= Egr ( M ') for any non zero graded η submodule M' ≤ M . Now let i be the inclusion mappings 3. M is graded uniform. Pure and Applied Mathematics Journal 2015; 4(2): 47-51 51

4. M= Egr ( U ) for some graded uniform module U. version of the important well known theorem of Matlis which states that : If R is either a commutative noetherian ring or a 5. M= E( Ru / ) for some graded meet-irreducible ideal gr right artinian ring then there exists a bijection between the set u≤gr R . of isomorphism classes of right indecomposable injective R-modules and the set of all prime ideals of R .On the other 6. M is strongly indecomposable, that is, Endgr ( M ) is a hand , one can also try to compute the graded injective hull of local graded ring. cyclic non-free graded modules and graded modules over Proof. 1) ⇒ 2) Direct consequence of Theorem 3.9 . graded noetherian rings which are not PID.Finally, the results 2) ⇒ 3) Clear, from the definition of graded uniform of this paper enable us to study the graded extended module . 3) ⇒ 4) Obvious. injectivities of graded modules such as Soc-injectivity , 4) ⇒ 5) Let V ≠ 0 be a cyclic submodule of U. Since U Rad-injectivity, and almost injectivity, cf. [1],[2],[4]. However this is not the aim of this paper, it will be the topic of some ≤ess . ≤ess . is graded uniform, Vgr U . But then Vgr E gr ( U ) by 1. forthcoming work. of Proposition 3.1, and this gives M= E( V ) . We are done gr by identifying V with R/ u for a (necessarily meet-irreducible) References right ideal u≤gr R . 5) ⇒ 6) Here M= Egr ( U ) where U= R/ u is uniform. [1] Amin I., Yousif M., Zeyada N., Soc-injective rings and α = modules, Comm. in Alg., Vol. 33,No.1, (2005), 4229-4250. If is a nonunit in E Endgr (R M ) , then kerα ≠ 0 . (If [2] Zeyada Nasr A., Hussein Salah El Din S., Amin Amr K., α = , then im(α ) ≤ M . But im(α ) ≅ M injective, ker 0 gr Rad-injective and almost-injective modules and rings, Alg. so M= im(α ) ⊕ A for some A ≠ 0 . Both im (α ) and A Colloquium, Vol.18, No. 3, (2011), 411-418. must intersect U nontrivially, contradicting the fact that U is [3] Nicholson W.K., Yousif M.F., Quasi-Frobenius rings, uniform.) Therefore, U I kerα ≠ 0 . If β is another nonunit Cambridge Tracts in Math. 158, Cambridge Univ. Press, in E, then likewise U I kerβ ≠ 0 and we have Cambridge, UK, (2003). [4] Yousif M. F., Zhou Y., Zeyada N., On peseudo-Frobenius rings, ker(αβ+ ) ⊇ (UI ker) α I ( U I ker β ) ≠ 0 . Can. Math. Bull., Vol.48 No. 2, (2005), 317-320.

This implies that α+ β is a nonunit in E; hence E is a [5] Nastasescu C., Van Oystaeyen F., Graded ring theory, Math. Library, North Holland, Amsterdam 28, (1982). local ring. 6) ⇒ 1) If E is a local ring, it does not have nontrivial [6] Nastasescu C., Van Oystaeyen F., Methods of graded rings, Springer-Verlag, Berlin Heidelberg, LNM.1836, (2004). idempotents [7] Kasch F., Modules and rings, Academic Press, New York, 4. Conclusion L.M.S. Monograph No.17, (1982). [8] Lam T. Y., Lectures on modules and rings, Graduate Texts in Along the proofs of the results of this paper one can easily Mathematics, Springer-Verlag, New York, Vol. 189, (1998). discover that the iterated use of essential extensions of graded modules seems to deserve further development. For instance, [9] Papp, Zolt’an, On algebraically closed modules, Publicationes using Theorem 3.11, it is quite possible to prove the graded Mathematicae Debrecen, Vol.6, (1959), 311-327.