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X-LIFTING MODULES OVER RIGHT PERFECT RINGS
X-LIFTING MODULES OVER RIGHT PERFECT RINGS
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2014. Apr, 21(2): 95-140
Copyright © 2014, Korean Society of Mathematical Education
  • Received : March 12, 2013
  • Accepted : April 02, 2014
  • Published : April 27, 2014
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About the Authors
Jong Moon Shin
DEPARTMENT OF MATHENATICS, DONGGUK UNIVERSITY, GYEONGJU 780-714, KoreaEmail address:shinjm@dongguk.ac.kr
Chae-hoon Chang
DEPARTMENT OF MATHEMATICS, DONGGUK UNIVERSITY, GYEONGJU 780-714, KoreaEmail address:yamaguchi21@hanmail.net

Abstract
Keskin and Harmanci defined the family B ( M,X ) = { A M | ∃ Y X , ∃ f ∈ Hom R ( M,X / Y ), Ker f / A M / A }. And Orhan and Keskin generalized projective modules via the class B ( M,X ). In this note we introduce X -local summands and X -hollow modules via the class B ( M,X ). Let R be a right perfect ring and let M be an X -lifting module. We prove that if every co-closed submodule of any projective module contains its radical, then M has an indecomposable decomposition. This result is a generalization of Kuratomi and Chang’s result [9, Theorem 3.4] . Let X be an R -module. We also prove that for an X -hollow module H such that every non-zero direct summand K of H with K B ( H,X ), if H H has the internal exchange property, then H has a local endomorphism ring.
Keywords
1. INTRODUCTION
Extending modules and lifting modules have been studied extensively in recent years by many ring theorists (see, for example, [3] , [5] - [14] ).
Let M and X be R -modules. In [8] , D. Keskin and A. Harmanci defined the family B ( M,X ) = { A M | ∃ Y X , ∃ f ∈ Hom R ( M , X / Y ), Ker f / A M / A }. They considered the following conditions:
  • B(M,X)-(D1): For anyA∈B(M,X), there exists a direct summandA* ≤⊕Msuch thatA/A* ≪M/A*
  • B(M,X)-(D2): For anyA∈B(M,X), ifB≤⊕M,M/A≃BimpliesA≤⊕M
  • B(M,X)-(D3): For anyA∈B(M,X) andB≤⊕M, ifA≤⊕MandM=A+BthenA∩B≤⊕M
They defined that M is said to be X - discrete if B ( M,X )-( D 1 ) and B ( M,X )-( D 2 ) hold, and is said to be X-quasi-discrete if B ( M,X )-( D 1 ) and B ( M,X )-( D 3 ) hold. Furthermore, M is said to be X-lifting if B ( M,X )-( D 1 ) holds. We have just seen that the following implications hold: X-discrete X-quasi-discrete X-lifting .
Throughout this paper, all rings R considered are associative rings with identity and all R -modules are unital.
Let M be a right R -module and N a submodule of M . The notation N M means that N is a direct summand of M .
A submodule K of M is called a small submodule (or superfluous submodule) of M , abbreviated K M , in the case when, for every submodule L M , K + L = M implies L = M .
2. PRELIMINARIES
Let A and P be submodules of M with P B ( M,X ). P is called an X-supplement of A if it is minimal with the property A + P = M equivalently, if M = A + P and A P P .
The module M is called X-amply supplemented if for any submodules A, B of M with A B ( M,X ) and M = A + B there exists an X -supplement P of A such that P B .
Let N 1 N 2 M . N 1 is a co-essential submodule of N 2 in M , abbreviated N 1 c N 2 in M , if the kernel of the canonical map M / N 1 M / N 2 → 0 is small in M / N 1 , or equivalently, if M = N 2 + X with N 1 X implies M = X .
A submodule N of M is said to be co-closed in M (or a co-closed submodule of M ), if N has no proper co-essential submodule in M . i.e., N ′ ≤ c N in M implies N = N ′. It is easy to see that any direct summand of a module M is co-closed in M . Note that every X -supplement submodule of M is co-closed in M .
For N ′ ≤ N M , N ′ is called a co-closure of N in M if N ′ is a co-closed submodule of M with N ′ ≤ c N in M . Any submodule of a module has a closure, however, co-closure does not exist in general.
Lemma 2.1 ( [9, Lemma 1.4] , [5, 3.2, 3.7] ). Let A B M . Then the following hold:
  • (1)A≤cB in M if and only if M=A+K for any submodule K of M with M=B+K.
  • (2) IfA≪M and B is co-closed in M, A≪B.
Lemma 2.2 (cf., [16, 41.14] ). Any projective module satisfies the following condition:
( D ) If M 1 and M 2 are direct summands of M such that M 1 M 2 M and M = M 1 + M 2 , then M = M 1 M 2 .
Lemma 2.3 ( [13, Theorem 3.5] ). If M is a lifting module with the condition (D), then M can be expressed as a direct sum of hollow modules .
Lemma 2.4 ( [1, Lemma 17.17] ). Suppose that M has a projective cover. If P is projective with an epimorphism 𝜑 : P M , then P has a decomposition P = P 1 P 2 such that P 1 Ker 𝜑 and 𝜑 | P2 : P 2 M is a projective cover of M .
Theorem 2.5 ( [3, Theorem 1.1.24] ). For an R-module M, the following hold:
  • (1)If M is a quasi-injective module, then M is a fully invariant submodule of E(M).
  • (2)If M is a quasi-injective module, then any direct decomposition E(M) =E1⊕ ⋯ ⊕Eninduces M= (M∩E1) ⊕ ⋯ ⊕ (M∩En).
  • (3)If M is a quasi-projective module with a projective cover𝜑 :P→M,Ker𝜑is a fully invariant submodule of P; whence any endomorphism of P induces an endomorphism of M.
  • (4)If M is a quasi-projective module with a projective cover𝜑 :P→M,then any direct decomposition P=P1⊕⋯⊕Pninduces M= 𝜑(P1)⊕⋯⊕𝜑(Pn).
A ring R is called right perfect if every right R -module has a projective cover.
Proposition 2.6. The following statements are equivalent:
  • (i)Every cyclic right R-module has a projective cover;
  • (ii)RRis a lifting module.
Proof . (i) ⟹ (ii) Let A be a submodule of RR and let 𝜑 : R R / A be the canonical epimorphism. Since R / A has a projective cover, by Lemma 2.4, there exists a decomposition RR = eR ⊕ (1 - e ) R such that (𝜑 | eR ) : eR R / A → 0 a projective cover and (1 - e ) R A . This implies Ker (𝜑 | eR ) = A eR eR . i.e., R = eR ⊕ (1 - e ) R such that A eR eR . Thus RR is lifting.
(ii) ⟹ (i) Suppose that RR is lifting. We claim that R / A has a projective cover. Since RR is lifting, for any A R , there exists A * ≤ c A such that R = A * ⊕ A **. Then 𝜋 | A** : A ** → R / A → 0 is a projective cover of R / A , where 𝜋 : R R / A → 0 is the canonical epimorphism. □
As corollaries of Proposition 2.6, we obtain the following two results.
Corollary 2.7. Let P be a projective module. Then the following statements are equivalent:
  • (i)Every factor module of P has a projective cover;
  • (ii)P is lifting.
Corollary 2.8. The following statements are equivalent:
  • (i)Every simple right R-module has a projective cover;
  • (ii)RRsatisfies the lifting property for simple factor modules.
Lemma 2.9 ( [9, Lemma 3.1] , [5, 3.2] ). Let f : M N be an epimorphism . Suppose K c K in M . Then f ( K ) ≤ c f ( K ′) in N .
Lemma 2.10 ( [8, Lemma 2.2] ). Let M, N and X be R-modules. Then the following hold:
  • (1)If A∈B(M,X)and B≤A with A/B≪M/B, then B∈B(M,X).
  • (2)Let h:M→N be an epimorphism and A∈B(M,X)with Ker h≤A.Then h(A) ∈B(N,X).Conversely, if h(A) ∈B(N,X)and Ker h≤A,then A∈B(M,X).
  • (3)Let B≤A≤M.Then A∈B(M,X)if and only if A/B∈B(M/B,X).
  • (4)Let h:N→M be an epimorphism and A∈B(M,X).Then h-1(A) ∈B(N,X).
3. MAIN RESULTS
Theorem 3.1. Let R be a ring. The following conditions are equivalent:
  • (1)R is right perfect;
  • (2)Every projective right R-module is lifting;
  • (3)Every quasi-projective right R-module is lifting;
  • (4)Every countably generated free right R-module is lifting.
Proof . (1) ⟺ (2) This follows from Corollary 2.7.
(2) ⟹ (3) Let QR be a quasi-projective module and let A be a submodule of Q . Consider the canonical epimorphism f : Q Q / A . We can take a projective module PR such that Q is a homomorphic image of P , i.e., we have an epimorphism g : P Q . Since P is a lifting module, by Lemma 2.4, there exists a decomposition P = P 1 P 2 such that P 1 g -1 ( A ), fg | P2 : P 2 Q / A is a projective cover. As Q is a quasi-projective module, the decomposition P = P 1 P 2 induces a direct decomposition Q = g ( P 1 )⊕ g ( P 2 ) by Theorem 2.5. Then g ( P 1 ) ≤ A and g ( P 2 ) ∩ A g ( P 2 ) hold.
(3) ⟹ (2) Obvious.
(1) ⟹ (4) This follows from [1, Theorem 28.4] .
(4) ⟹) (1) By (4), R is semiperfect and R / J ( R ) is semisimple. Since R (
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) is lifting, there exists a decomposition R (
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) = X Y such that X ≤ Rad( R (
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) ) and Rad( R (
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) ) ∩ Y Y . Because Rad( R (
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) ) = Rad( X )⊕Rad( Y ) and X ≤ Rad( R (
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) ), we see Rad( X ) = X , which implies X = 0 and R (
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) J ( R ) = Rad( R (
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) ) R (
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) . Hence, by [1, Lemma 28.3] , J ( R ) is right T -nilpotent. Thus R is right perfect. □
A family { X λ | λ ∈ Λ} of submodules of a module M with X λ B ( M,X ) is called an X-local summand of M , if ∑ λ∈Λ X λ is direct and ∑ ⊕ λ∈F X λ M for every finite subset F ⊆ Λ.
By anology with the proof of [14, Lemma 2.4] or [11, Theorem 2.17] , we have the following lemma.
Lemma 3.2. If every X-local summand of a module M is a direct summand, then M has an indecomposable decomposition.
By Lemma 2.1(1), we have the following lemma.
Lemma 3.3. Assume Pi c Qi in P for every i I . Then ∑⊕ iI Pi c ∑⊕ iI Qi in P.
Lemma 3.4. Let { Pi } iI be a set of R-modules. Assume Pi B ( M,X ) for every i I . Then ∑⊕ iI Pi B ( M,X ).
Proof . Since Pi B ( M,X ), there exist a submodule Y of X and a homomorphism fi : M X / Y such that Ker f / Pi M / Pi . Put f = ∑⊕ iI fi . Then f : M X / Y such that
Ker f/∑⊕iI PiM/∑⊕iI Pi.
Thus ∑⊕ iI Pi B ( M,X ).           □
Lemma 3.5. Let X be a right R-module. Suppose that R is a right perfect ring. Then every projective right R-module is X-lifting.
Proof . Let P be a projective module. For any A B ( P,X ), consider the canonical epimorphism 𝜑 : P P / A . Since P / A has a projective cover, by Lemma 2.4, there exists a decomposition P = P 1 P 2 such that P 1 ≤ Ker 𝜑 and 𝜑 | P2 : P 2 P / A is a projective cover of P / A . Hence P is X -lifting.          □
Proposition 3.6. Let R be a right perfect ring and let M be an X-lifting module. Then M is X-amply supplemented.
Proof . Let A , B M such that B B ( M,X ) and M = A + B . Since M = A + B and B B ( M,X ), there exist Y X and f : M X / Y such that Ker f / B M / B .
Consider the isomorphism 𝛼 : M / B A / A B . Then 𝛼(Ker f/B) = Ker f / A B . Hence Ker f / A B M / A B . Therefore A B B ( M,X ). As M is X -lifting, there exists a direct summand K of M such that K c A B in M . Then A B = K ⊕ [ K * ∩ ( A B )], M = ( A B ) + K * and ( A B ) ∩ K * ≪ K *. Thus M = B + ( A K *).
Let D be a co-closure of A K * in M . Then M = B + D and B D B ∩ ( A K *) ≪ K *. Hence B D K *. Since D is co-closed in M , B D D and B D M , B D D . Thus D is an X -supplement of B in M such that D A .          □
Lemma 3.7 ( [8, Lemma 3.2] ). Every epimorphic image of an X-amply supplemented R-module is X-amply supplemented.
Lemma 3.8. Let M be an X-amply supplemented module and let Ker f M
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N → 0. Suppose K is co-closed in M with Ker f K . Then f ( K ) is co-closed in N .
Proof . By Lemma 3.7, N is X -amply supplemented. Let L c f ( K ) in N . We claim that L = f ( K ). Since f is an epimorphism, there exists a submodule T of K in M with f ( T ) = L . Since N is X -amply supplemented, there exists an X -supplement P of f ( K ) such that P N . i.e., N = f ( K ) + P and f ( K ) ∩ P P . Since f is an epimorphism, there exists a submodule Q of M with f ( Q ) = P . Then M = K + Q +Ker f . As Ker f M, M = K + Q . This implies N = f ( K )+ f ( Q ) = f ( K )+ P = L + P = f ( T )+ f ( Q ). Then M = T + Q +Ker f = T + Q . Thus T c K in M by Lemma 2.1(1). As K is co-closed in M, T = K . Hence L = f ( T ) = f ( K ). Therefore f ( K ) is co-closed in N .           □
Proposition 3.9. Suppose that M is an X-lifting module. Then every co-closed submodule K of M with K B ( M,X ) is a direct summand .
Proof . Since M is X -lifting, there exists a direct summand K * such that K * ≤ c K in M . As K is co-closed in M, K = K * ≤ M .          □
Theorem 3.10. Let R be a right perfect ring and let M be an X-lifting module. Assume that every co-closed submodule of any projective module contains its radical. Then every X-local summand of M is a direct summand .
Proof . Let M be an X -lifting module and let ∑ iI Xi be an X -local summand of M with Xi B ( M,X ). Since R is right perfect, M has a projective cover, say Ker f P
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M → 0. By Lemma 3.5, P is projective X -lifting. Since Xi B ( M,X ), f -1 ( Xi ) ∈ B ( P,X ) by Lemma 2.10(4). So there exists a decomposition P = Pi
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( i I ) such that Pi c f -1 ( Xi ) in P . By Lemma 2.9, f ( Pi ) ≤ c f ( f -1 ( Xi )) = Xi in M . As Xi is co-closed in M , f ( Pi ) = Xi . First we prove that ∑ iI Pi is direct. Let F be a finite subset of I - { i }. Since ∑⊕ iI Xi is an X -local summand of M , we see
f(Pi + ∑jFPj) = Xi ⊕ (∑ ⊕jF Xj) ≤ M.
So there exists a direct summand Y of M such that M = Xi ⊕ (∑ ⊕ jF Xj ) ⊕ Y . As P is lifting, there exists a decomposition P = Q Q * such that Q c f -1 ( Y ) in P . Then f ( Q ) = Y . Thus we see
P = Pi + ∑jFPj + Q + Ker f = Pi + ∑jFPj + Q.
Then Pi ∩(∑ jF Pj + Q ) ⊆ Ker f P . Similarly, we see Q ∩( Pi +∑ jF Pj P and Pj ∩ ( Pi +∑ lF-{j} Pl + Q ) ≪ P . By Lemma 2.2, we obtain P = Pi ⊕ (∑ jF Pj ) ⊕ Q . Hence ∑ iI Pi is direct. By the same argument, we see ∑ ⊕ iI Pi is an X -local summand of P . By Lemma 2.3, ∑ ⊕ iI Pi P . So f (∑ ⊕ iI Pi ) is co-closed in M by Lemma 3.8. Since M is X -lifting, we see
∑ ⊕iI Xi = f(∑ ⊕iI Pi) ≤ M.
Thus any X -local summand of M is a direct summand. □
By Lemma 3.2. and Theorem 3.10, we obtain the first main theorem.
Theorem 3.11. Suppose that every co-closed submodule of any projective module contains its radical. Then every X-lifting module over right perfect rings has an indecomposable decomposition.
Let X be an R -module. A non-zero R -module H is X-hollow if for any proper submodule K of H with K B ( H,X ), K H .
Proposition 3.12. Let H and X be R-modules. Assume that every non-zero direct summand K of H with K B ( H,X ). Then H is X-hollow if and only if H is indecomposable X-lifting.
Proof. (⟹) Assume H is X -hollow. Let K B ( H,X ) with K H . Since H is X -hollow, K H . So there exists a decomposition H = 0 ⊕ H such that 0 ≤ c K in H . Thus H is X -lifting. Now, assume that H = H 1 H 2 , Hi ≠ 0, i = 1, 2. Since H is X -hollow, Hi H , i = 1, 2. Hence Hi = 0. This is a contradiction. Therefore H is indecomposable.
(⟸) Suppose that H is indecomposable X -lifting. Let K B ( H,X ) with K H . By hypothesis, there exists a decomposition H = K * ⊕ K ** such that K * ≤ c K in H . As H is indecomposable, we have either K * = 0 or K ** = 0. If K * = 0, then K H . In the second case, H = K . This is a contradiction. □
A module M is said to have the ( finite ) exchange property if, for any (finite) index set I , whenever M N = ⊕ iI Ai for modules N and Ai , then M N = M ⊕(⊕ iI Bi ) for some submodules Bi Ai . A module M has the ( finite ) internal exchange property if, for any (finite) direct sum decomposition M = ⊕ iI Mi and any direct summand X of M , there exist submodules
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Mi such that M = X ⊕ (⊕ iI
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).
By Proposition 3.12, we obtain the second main theorem.
Theorem 3.13 (cf., [15, Proposition 1] ). Let X be an R-module and let H be an X-hollow module. Assume that every non-zero direct summand K of H with K B ( H,X ). If H H has the internal exchange property, then H has a local endomorphism ring.
Corollary 3.14 (cf., [5, 12.2] ). Let X be an R-module and let H be an X-hollow module. Assume that every non-zero direct summand K of H with K B ( H,X ). Then the following conditions are equivalent:
  • (1)H has a local endomorphism ring;
  • (2)H has the finite exchange property;
  • (3)H has the exchange property.
References
Anderson F.W. , Fuller K.R. 1992 Rings and Categories of Modules Springer-Verlag Berlin-Heidelberg-New York
Azumaya G. , Mbuntum F. , Varadarajan K. 1975 On M-projective and M-injective modules Pacific J. Math. 95 9 - 16
Baba Y. , Oshiro K. Artinian Rings and Related Topics, Lecture Note
Bass H. 1960 Finitistic dimension and homological generalization of semiprimary rings Trans. Amer. Math. Soc. 95 466 - 486    DOI : 10.1090/S0002-9947-1960-0157984-8
Clark J. , Lomp C. , Vanaja N. , Wisbauer R. 2007 Lifting modules Birkhauser Boston Boston
Dung N.V. , Huynh N.V. , Smith P.F. , Wisbauer R. 1994 Extending modules Longman Group Limited
Harada M. 1983 Factor categories with applications to direct decomposition of modules Marcel Dekker New York
Keskin D. , Harmanci A. 2004 A relative version of the lifting property of modules Algebra Colloquium 11 (3) 361 - 370
Kuratomi Y. , Chang C. Lifting modules over right perfect rings Comm. Algebra
López-Permouth S.R. , Oshiro K. , Rizvi S.T. 1998 On the relative (quasi-) continuity of modules Comm. Algebra 26 (11) 3497 - 3510    DOI : 10.1080/00927879808826355
Mohamed S.H. , Müller B.J. 1990 Continuous and Discrete modules London Math. Soc. Lect. Notes Cambrige Univ. Press 147
Orhan N. , Keskin D. 2005 Characterizations of lifting modules in terms of cojective modules and the class of B(M,X) Int. J. Math. 16 (6) 647 - 660    DOI : 10.1142/S0129167X05003041
Oshiro K. 1983 Semiperfect modules and quasi-semiperfect modules Osaka J. Math. 20 337 - 372
Oshiro K. 1984 Lifting, extending modules and their applications to QF-rings Hokkaido Math. 13 310 - 338    DOI : 10.14492/hokmj/1381757705
Warfield R.B. 1969 A Krull-Schmidt theorem for infinite sums of modules Proc. Amer. Math. Soc. 22 460 - 465    DOI : 10.1090/S0002-9939-1969-0242886-2
Wisbauer R. 1991 Foundations of Modules and Ring Theory Gordon and Breach Science Publishers