Let R be an associative ring with identity and let M be an unitary  left R-module. An R-module M is called F1- lifting, if every fully invariant sub module A of M contains a direct summand  B of M such that B ≤ ce A . In this paper we introduce  F1-δ-lifting  as a generalization of F1-lifting  module. We prove similar results of  F1-lifting. Keywords:  F1-δ-lifting modules ,strongly F1-δ-lifting modules

Hasan, Aseel Amer

Hasan, Wasan Khalid

English

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Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.7, 2016  1 F1- Delta -Lifting modules  1  Wasan Khalid  Hasan,  Aseel Amer Hasan 2 1Departement of Mathematics, College of Science, Baghdad University, Baghdad, Iraq, 2Departement of Mathematics, College of Science, Baghdad University, Baghdad , Iraq.  Abstract:                 Let R be an associative ring with identity and let M be an unitary  left R-module. An R-module M is called F1- lifting, if every fully invariant sub module A of M contains a direct summand  B of M such that B ≤ ce A . In this paper we introduce  F1-δ-lifting  as a generalization of F1-lifting  module. We prove similar results of  F1-lifting. Keywords:  F1-δ-lifting modules ,strongly F1-δ-lifting modules. Introduction         Let R be an associative ring with identity, and let M be a unitary left R-module. A sub module N of an R-module M is called small in M denoted by ( N˂˂ M), if whenever M = N + K, for K ≤ M implies  K = M. [6]. We say that a sub module K is a coessential sub module of N in M ( denoted by K ≤ ce N ), if N / K ˂˂ M / K,[6]. A module M is called lifting or satisfies (D1) , if every sub module N of M contains a direct summand K of M  such that K ≤  ce N.[4] . A non- zero module M is called hollow, if every proper sub module of M is small in M, [3]. Recall that Z(M) = { mϵ M; ann(m) ≤  e R}, is the singular sub module of M, if Z(M) = M then M is singular , and if Z(M) = 0, then M is non- singular. A sub module N of M is called δ-small in M, if whenever M = N + K,   K ≤ M with M / K singular implies M = K [6]. A non- zero module M is called δ- hollow, if every proper sub module of M is   δ- small. It is clear that every hollow module is δ-hollow  , but the converse in general is not hold. Consider R is a semi simple ring and M is a non- zero R-module. Then M is non-singular and semi simple.For any non- zero sub module N ≤  M, N is a direct summand of M , and hence is not small in M, but every sub module of M (even M itself)        Is δ-small. A sub module N of M is called fully invariant, if  f(N) ≤ N for every f ϵ EndR(M). In [5] it was introduced F1-lifting module. An R-module M is called F1-lifting, if every fully invariant sub module N of M contains a direct summand K of M such that K≤ ce N .According to this definition we introduce another generalization for lifting modules. We introduce F1- δ-lifting module. A Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.7, 2016  2 module M is called F1-δ-lifting, if every invariant sub module N of M contains a direct summand K of M such that N / K ˂˂ δ M / K and we prove similar results of F1-lifting module.[5]. §1 F1-δ- lifting module           In this section we introduce F1-δ- lifting module and give some properties of this type of module, but first we recall some known results which will be needed in our work. Lemma(1.1) :-[1.lemma 1.1] Let M be an R-module, then  1. Any sum of fully invariant sub module of M is again a fully invariant sub module of M. 2. Any intersection of fully invariant sub module is again a fully invariant sub module of M. 3.  If  X ≤ Y≤ M, such that  Y is a fully invariant sub module of M and X is a fully invariant sub module of Y, then X is a fully invariant sub module of M . 4. If  M = ⊕iϵ I Xi and S is a fully invariant sub module of M, then     S = ⊕iϵ I ( Xi ∩ S ) and  (Xi ∩ S ) is a fully invariant sub module of  Xi ,         ⩝ i ϵ I. Definition (1.2) :  Let N , K be sub module of an R - module M such that K ≤ N, we say that K is a generalized coessential sub module of  N ( denoted by K ≤  δce N ), if N / K ˂˂ δ M / K .            It is clear that if K ≤ ce N , then K ≤ δce N, but the convers in general is not true , if R is semi simple ring and M is non-zero R- module, then M is semi simple and non-singular hence ⩝ N ≤ M , 0 ≤ N is not small in M / 0, but N / 0 ˂˂δ M / 0.  Definition(1.3): Let M be an R-module, we say that M is F1-δ-lifting, if every fully invariant sub module N of M contains a direct summand K such that K ≤ δce N.       It is clear that every δ- hollow module is F1-δ-lifting.       By [5]. M is F1-δ-lifting, if and only if every fully invariant A of M can be written as A = B ⊕ S where B is a direct summand of M and    S˂˂ M.       In the following we prove a similar result for F1-δ-lifting module. Proposition(1.4):- The following are equivalent for an R-module M. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.7, 2016  3 1. M is F1-δ-lifting 2. Every fully invariant sub module A of M can be written as  A = B⊕ S, where B is a direct summand of M and S˂˂δ M.  3. Every fully invariant sub module A of M can be written as  A = B⊕ S, where B is direct summand of M and S ˂˂δ M . Proof:   1 → 2 )  Let A be a fully invariant sub module of M, then by (1) , ∃ B ≤ M such that A / B ˂˂δ  M / B, hence ∃ K ≤ M   such that  M=  B ⊕ K. Then             A = A∩ B⊕  A∩ K = B⊕ K ∩ A  , take S = K ∩ A.          Now let M = K ∩ A + K' with M / K' singular , Hence   M / B = ( K ∩ A + B) / B + K' / B= A / B + K' / B.  M / B / K' / B ≅ 𝑀/ K' singular, and A / B ˂˂δ M / B then M = K'.  2→ 3) clear  3→ 1) Let A be a fully invariant sub module of M then by (3) A = B + S, where B ≤⊕ M and S ˂˂δ M. Let M = B⊕ C for C ≤ M and let  M / B = A / B + K / B, where (M / B) / ( K / B ) ≅ M / K singular then  M= A + K = A + B +S = K + S. Since M / K singular and S˂˂δ M, then       M = K. hence A / B ˂˂δ M / B. Theorem( 1.5):  Let M = M1 ⊕ M2  be a direct summand of F1- δ- lifting modules, then M is F1-δ- lifting. Proof:  Let A be a fully invariant sub module of M, then A = A ∩ M1⊕ A ∩ M2, and A ∩ Mi is fully invariant ⩝ i = 1,2 , since Mi is F1-δ- lifting ,i=1,2, then by pro.(1.4) , A ∩Mi = Bi ⊕ Si, where Bi is a direct summand of Mi,  Si ˂˂δ Mi , ⩝ i = 1,2…         Let B = B1 ⊕ B2, and S = S1⊕ S2, then  A = B⊕ S where B is a direct summand of M and S ˂˂δ M. Corollary(1.6): Let M = M1⊕ M2⊕…⊕ Mn be a direct sum of F1 -δ -lifting module, then M is F1-δ-  lifting. Corollary (1.7): If M is a finite direct sum of δ- hollow modules, then M is F1-δ -lifting.  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.7, 2016  4 Remark(1.8):  It is clear that every lifting module isF1-δ -lifting , but the convers in general is not true, for example, consider Z - module  M = Z / pZ  ⊕ Z / p3Z each Z/ pZ , Z / p3Z is hollow, hence F1- δ - lifting therefore by  theorem(1.5), M is F1-δ-lifting, but not lifting,[2].      Recall that a pair(f,P) is called a projective δ - cover for an R- module M, if P is projective and f: P → M is an epimorphisim  with  Kerf ˂˂δ P, [7].  Proposition (1.9):  Let P be  a projective R-module ,if P is F1-  δ -lifting, then P / A has a projective δ- cover. For every fully sub module A of B.        The convers holds, if for every fully invariant sub module A of P, P / A has a projective δ- cover . f: X → P /A , such that X is uniform. Proof: Let P be a projective F1-δ- lifting module and let A be a fully invariant sub module of P, then ∃ B ≤ A, B ≤ ⊕ P such that A = B ⊕  S , S˂˂δ P, hence P = B ⊕C, for C ≤ P , then( B + S) / B ˂˂δ P / B, thus      𝜋: P / B→ P / ( B + S) = P / A , P / B is projective , and  Ker 𝜋 = { w ∈ P/B ; 𝜋(w) = A }= { x ∈ B ; x + A = A } = A / B then  Ker 𝜋  = A / B ˂˂δ P/ B. ←))  Let A be a fully invariant sub module of P, let f : X → P / A be a projective δ -cover for P / A such that  X is uniform. Consider the following diagram.                                                                         P                                                            h                π                                           X                            P/A                  0      Since P is projective, then ∃ h : P → X , such that f ° h = 𝜋 .Let x ∈ X, then f(x) ∈ P / A. Since 𝜋 is an epimorphisim, then ∃  y  ∈ P, such that    𝜋(y) = f(x),thus f °h(y) = f(x), hence x – h(y) ∈ Kerf, therefore X = Kerf + h(p). Since Kerf ˂˂ δ X and X is uniform, then by [7], Ker f˂˂ X. Hence h(P) = X,  then h: P→ X is an epimorphisim thus h splits , therefore P = Ker h ⊕  K for K ≤ P, and A = Ker h ⊕ K∩ A. where  Ker h ≤  ⊕ P and K ∩ A ˂˂δ P.   f Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.7, 2016  5 § 2 Strongly F1- δ – lifting modules.          In this section we introduce a strongly F1-δ- lifting .According to the definition that appeared in [5]. And we prove some results on this type of modules similar to results of strongly F1-δ- lifting module [5].  Definition(2.1): Let M be an R-module. We say that M is strongly F1-δ-lifting , if every fully invariant sub module A of M contains a fully direct summand B of M such that B ≤ Gce A. Proposition(2.2):  For any R- module M , then following are equivalent: 1. M is a strongly F1-δ-lifting module . 2. Every fully invariant sub module A of M can be written as  A = B⊕ S, where B is fully invariant direct summand of M and     S˂˂ δ M. 3. Every fully invariant sub module A of M can be written as  A = B + S, where B is a fully invariant direct of M and S ˂˂ δM. Proposition(2.3): Let M be an F1- δ-lifting module , if 0 is the only δ- small sub module of  M, then every fully invariant sub module of M is strongly F1-δ – lifting module. Proof: Let N be a fully invariant sub module of M and let A a fully invariant sub module of N, then A is fully invariant in M [ 1,Lemma 1.1].Since M is F1-δ-lifting , then A = B ⊕  S, where  B ≤ ⊕ M and S ˂˂δ M. hence  S = 0, thus A ≤ ⊕ M therefore N is strongly F1- δ- lifting module. Theorem(2.4): A direct summand of a strongly F1-δ- lifting is again strongly F1-δ – strongly F1-δ- liftig. Proof: Let M = W ⊕ V , be a strongly F1- δ- lifting , suppose that A is a fully invariant sub module of W then ∃ A2 a fully invariant sub module of V. Such that A1⊕ A2 is fully invariant in M see[ 6 ]. since M is strongly F1- δ- lifting then A1⊕ A2= B⊕ S, where B ≤ ⊕ M, B isa fully invariant sub module of M and S ˂˂δ M, then B = ( B ∩ W)  ⊕ (B ∩ V ) and B ∩ W is fully invariant in W also B ∩ W is a direct summand of M . Let π1: M → W and π2:M → V then A1 = π1(B) + π1(S)= W ∩ B + π1(S) since S ˂˂δ M, π1(S) ˂˂ δ W by  [1], therefore W is F1-δ - lifting. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.6, No.7, 2016  6 Proposition(2.5):Let M =⊕i=1 Mi, and let Mi be a fully invariant     sub module of M ,∀ i= 1,…n, then M is strongly F1- δ -lifting if and only if Mi is strongly F1-δ - lifting,∀ i= 1,…n. Proof: →) By prop(3.4)  ←) Suppose that M = ⊕i=1 Mi  and Mi is fully invariant ∀ i= 1,…n.      Let N be a fully invariant sub module of M then  N= ⊕i=1( N ∩ Mi) and N ∩ Mi is fully invariant,∀ i= 1,…n., since Mi is strongly F1-δ-lifting , then N ∩ Mi = Bi ⊕ Si , where Bi is fully invariant direct summand of Mi and Si˂˂δ Mi ,∀ i= 1,…n. Let  B =⊕i=1 Bi and S=⊕i=1 Si then B = B + S , where B ≤ ⊕ M and S˂˂ δ M.  REFERNCES [1] G.F. Birknmeier, B. J. Muller and S. T. Rizvi, Modules in which every fully invariant sub module is essential in direct summand , Comm. Algebra Vol.30 (2002), 1395-1415. [2] J. Clark, C. lomp, N. Vanaja and R. Wisbauer, lifting modules, Frontiers in Mathematics, Birkhauser,(2006). [3] P.Fleury. Hollow modules and local Endomorphisim Rings, Pac. J. Math. , 53(1974),pp.379-385. [4] D. Keskin, On lifting modules. Comm. Algebra 28 (7)(2000), 3427-3440. [5] Y. Talebi and Amoozegar ; strongly F1-δ- lifting modules, International Electronic Journal of Algebra, V3 (2003), 75-82. [6] R. Wisbauer , Foundations of modules and Ring , Gordan and Breach  Reading ,1991. [7] Y.Zhou, Generalization of perfect semi perfect and semi-regular rings, Algebra Colloquim , 7(3) (2000),pp.305 – 318.                  

F1- Delta -Lifting modules

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F1- Delta -Lifting modules

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