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. 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: 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 . 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: 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: A ring R is called right perfect if every right R -module has a projective cover. Proposition 2.6. The following statements are equivalent: Proof . (i) ⟹ (ii) Let A be a submodule of R_R 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 R_R = 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 R_R is lifting. (ii) ⟹ (i) Suppose that R_R is lifting. We claim that R / A has a projective cover. Since R_R 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: Corollary 2.8. The following statements are equivalent: 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: Theorem 3.1. Let R be a ring. The following conditions are equivalent: Proof . (1) ⟺ (2) This follows from Corollary 2.7. (2) ⟹ (3) Let Q_R 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 P_R 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 ^{( )} is lifting, there exists a decomposition R ^{( )} = X ⊕ Y such that X ≤ Rad( R ^{( )} ) and Rad( R ^{( )} ) ∩ Y ≪ Y . Because Rad( R ^{( )} ) = Rad( X )⊕Rad( Y ) and X ≤ Rad( R ^{( )} ), we see Rad( X ) = X , which implies X = 0 and R ^{( )} J ( R ) = Rad( R ^{( ) )} ≪ R ^{( )} . 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 P_i ≤ _c Q_i in P for every i ∈ I . Then ∑⊕ _i∈I P_i ≤ _c ∑⊕ _i∈I Q_i in P. Lemma 3.4. Let { P_i } _i∈I be a set of R-modules. Assume P_i ∈ B ( M,X ) for every i ∈ I . Then ∑⊕ _i∈I P_i ∈ B ( M,X ). Proof . Since P_i ∈ B ( M,X ), there exist a submodule Y of X and a homomorphism f_i : M → X / Y such that Ker f / P_i ≪ M / P_i . Put f = ∑⊕ _i∈I f_i . Then f : M → X / Y such that Ker f/∑⊕_i∈I P_i ≪ M/∑⊕_i∈I P_i. Thus ∑⊕ _i∈I P_i ∈ 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 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 ∑ _i∈I X_i be an X -local summand of M with X_i ∈ B ( M,X ). Since R is right perfect, M has a projective cover, say Ker f ≪ P M → 0. By Lemma 3.5, P is projective X -lifting. Since X_i ∈ B ( M,X ), f ^-1 ( X_i ) ∈ B ( P,X ) by Lemma 2.10(4). So there exists a decomposition P = P_i ⊕ ( i ∈ I ) such that P_i ≤ _c f ^-1 ( X_i ) in P . By Lemma 2.9, f ( P_i ) ≤ _c f ( f ^-1 ( X_i )) = X_i in M . As X_i is co-closed in M , f ( P_i ) = X_i . First we prove that ∑ _i∈I P_i is direct. Let F be a finite subset of I - { i }. Since ∑⊕ _i∈I X_i is an X -local summand of M , we see f(P_i + ∑_j∈FP_j) = X_i ⊕ (∑ ⊕_j∈F X_j) ≤_⊕ M. So there exists a direct summand Y of M such that M = X_i ⊕ (∑ ⊕ _j∈F X_j ) ⊕ 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 = P_i + ∑_j∈FP_j + Q + Ker f = P_i + ∑_j∈FP_j + Q. Then P_i ∩(∑ _j∈F P_j + Q ) ⊆ Ker f ≪ P . Similarly, we see Q ∩( P_i +∑ _j∈F P_j ≪ P and P_j ∩ ( P_i +∑ _l∈F-{j} P_l + Q ) ≪ P . By Lemma 2.2, we obtain P = P_i ⊕ (∑ _j∈F P_j ) ⊕ Q . Hence ∑ _i∈I P_i is direct. By the same argument, we see ∑ ⊕ _i∈I P_i is an X -local summand of P . By Lemma 2.3, ∑ ⊕ _i∈I P_i ≤ _⊕ P . So f (∑ ⊕ _i∈I P_i ) is co-closed in M by Lemma 3.8. Since M is X -lifting, we see ∑ ⊕_i∈I X_i = f(∑ ⊕_i∈I P_i) ≤_⊕ 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 , H_i ≠ 0, i = 1, 2. Since H is X -hollow, H_i ≪ H , i = 1, 2. Hence H_i = 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 = ⊕ _i∈I A_i for modules N and A_i , then M ⊕ N = M ⊕(⊕ _i∈I B_i ) for some submodules B_i ≤ A_i . A module M has the ( finite ) internal exchange property if, for any (finite) direct sum decomposition M = ⊕ _i∈I M_i and any direct summand X of M , there exist submodules ≤ M_i such that M = X ⊕ (⊕ _i∈I ). 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: