Some Results on C -retractable Modules

. An R -module M is called c -retractable if there exists a nonzero homomorphism from M to any of its nonzero complement submodules. In this paper, we provide some new results of c retractable modules. It is shown that every projective module over a right SI -ring is c -retractable. A dual Baer c -retractable module is a direct sum of a Z 2 -torsion module and a module which is a direct sum of nonsingular uniform quasi-Baer modules whose endomorphism rings are semi-local quasi-Baer. Conditions are found under which, a c -retractable module is extending, quasi-continuous, quasi-injective and retractable. Also, it is shown that a locally noetherian c -retractable module is homo-related to a direct sum of uniform modules. Finally, rings over which every c retractable is a C 4 -module are determined.


Introduction
Throughout all rings are associative with identity and all modules are unitary right module. Let R be a ring. Following [19], we say that an R-module M is retractable if Hom R (M, N ) = {0} for any nonzero submodules N of M . It is shown in [19] that every projective module over a right V -ring is retractable. In [19] again, the semisimplicity of retractable modules is studied. M. R. Vedadi [23], introduced the concept of essentially retractable modules and proved that over semiprime right nonsingular rings, a nonsingular essentially retractable module is precisely a module with non-zero dual. In [7], A. Ghorbani and M. R. Vedadi introduced and studied the notion of epi-retractable module, where a module M is called epi-retractable if every submodule of M is a homomorphic image of M . They reveal some applications of projective, nonsingular, injective epi-retractable modules regarding the characterization of Bezout, pri, quasi-Frobenius rings. Note that epi-retractable modules are retractable. Earlier, P. F. Smith and A. Tercan [20] introduced C 11 -module as a generalization of extending modules, where a module M is said to be satisfy C 11 -condition if every submodule of M has a complement which is a direct summand. It is shown in ( [20], Theorem 2.7) that a module satisfies (C 11 if and only if M = Z 2 (M ) ⊕ K for some (nonsingular) K of M and Z 2 (M ) and K both satisfy (C 11 ). Later, the same authors investigated when a direct summand of a C 11 -module inherits the property [21]. Recently, t-closed submodules of a module M are defined in [2] as closed submodules of M which contain Z 2 (M ). In [3], S. H. Asgari, A. Haghany and A.R. Rezaei studied the modules M for which C 11 -condition holds for t-closed submodules (T 11 -type, for short). They showed among others the following results: i) A T 11 -type module is exactly a direct sum of a Z 2 -torsion module and a nonsingular C 11 -modules. (ii) A T + 11 -module (modules for which direct summands are T 11 -type) is precisely a direct sum of Z 2 -torsion and nonsingular C + 11 -module (modules for which direct summands satisly C 11 ). A. W. Chatters and S. M. Kheuri [4] defined the concept of c-retractable module, where an R-module M is called c-retractable if Hom R (M, C) = {0} for any nonzero complement submodules C of M . This notion is a generalization of both the retractable modules and the extending modules. They have shown that if M is a nonsingular c-retractable module such that S S is extending, then M is extending. But the converse is not true in general. On the other hand it is shown in [22] that if M is a retractable wd-Rickart module, then every indecomposable submodule of M is a simple direct summand. Motivated by the definition of the modules mentioned above and the results on retractable and c-retractable modules, we investigate the c-retractibility. Our aim in this paper is to give some new results on c-retractable modules. In general, c-retractable modules need not be projective and vice versa. Connections between projectivity and c-retractibility are investigated. Conditions are found under which, a c-retractable module is extending, quasi-continuous, quasi-injective and retractable. With the help of c-retractability, we investigated when the notions of K-nonsingularity and Baer modules are equivalent. Also, we characterize semisimple artinian rings in termes of c-retractable modules. Our paper is structured as follows: In the second section, we are going to give preliminary definitions which we will use throughout this paper. In the third section, we are going to show among others, the following results: (1) Every projective module over a right SI-ring is c-retractable.

Preliminaries
In this section, we are going to give preliminary definitions which we will use throughout this paper.
there is an essential submodule V ≤ e M that is a direct sum of n uniform submodules.

Main results
Remark 1. Cleary, every retractable module is c-retractable. The converse is not true in general. For example: Q as a Z-module is c-retractable while it is not retractable.. [4], Example 2.4).

Remark 3. ([4], Example 3.2)
Let R be the ring of all 2 by 2 upper triangular matrices which have arbitrary real numbers on the diagonal and an arbitrary complex number in the (1, 2)-position and let e ij be the element of R with 1 in the (i, j)-position and 0 elsewhere. Set P = e 11 R and let K denote the field of real numbers. Hence, P is a nonsingular projective R-module which is not a c-retractable R-module while the R-module M = R ⊕ P is c-retractable. This shows that a direct summand (hence a submodule or a factor module) of a c-retractable module need not be c-retractable.
Then, K ⊆ c M by Proposition 6.28 in [11]. Thus, there exists a nonzero homomor-

Proof.
Let N/C ⊆ c M/C. By the c-retractable condition on M , there is a nonzero homomorphism g : M −→ N . From this and by our assumption, Hom R (M/C, N/C) = 0. Proof.
It follows that (L⊕0) is a fully invariant complement submodule of M . Now, an application of Proposition 1 shows that N is c-retractable. Proof.
Proposition 5. If an arbitrary direct sum of copies of M is c-retracatable, then M is c-retractable.

Remark 4.
A projective module need not be c-retracatble and vice-versa. In fact a simple is c-retractable but not be projective. Moreover, by Remark 3, there is a projective module which is not c-retractable. In the following, we show that certains classes of projective modules are c-retractable.
Following [24], we call an R-module SI if every singular module is M -injective. Recall that a ring R is called right SI, if every singular R-module is injective. The following conditions are equivalent for a ring R.
(1) R is a right SI-ring.
(2) Every R-module is a SI-module.  Proof. This is clear. Proof. The suffiency follows from Lemma 3. Conversely, assume that M is any wd-Rickart cretractable module. Let 0 = C ⊆ c M . Since M is c-retractable, there is a nonzero endomorphism ϕ of M such that Imϕ ⊆ C. Thus, the wd-Rickart property of M implies that C contains a nonzero direct summand.

Proof.
Let M be any wd-Rickart c-retractable module. Let C be an indecompsable complement submodule of M . Let D any nonzero complement submodule of C. Since D ⊆ c M , we infer from Lemma 4 that D contains a nonzero direct summand E of M . As E ≤ C ≤ M and E ≤ ⊕ M , E ≤ ⊕ C. Since C is indecomposable, C = E = D. It follows that D is a direct summand of C, and hence C is an extending module. Since C is indecomposable, C is uniform.

Proof.
Suppose that M is a uniform-extending c-retractable module. Since local summands of M are summand, M is a direct sum of indecomposable modules (see [14], Theorem 2.  Proof. This follows from Theorem 3 and the fact that any local summand of a quasi-discrete module is a summand (see [6], Corollary 4.13).

Proof.
(1) ⇒ (2) Suppose M is ADS and c-retractable. Since M is dual Baer, we infer from Proposition 6(1) that M = ⊕ i∈I M i is a direct sum of uniform modules. Thus, every M i is quasi-continuous for every i ∈ I. On the other hand since M is ADS, we infer from Lemma 3.1 in [1] that ⊕ i =j∈I M j is M i -injective for every i ∈ I. Therefore M is quasi-continuous by ( [14], Theorem 2.13). Now, let ϕ be an essential monomorphism of M . Then Imϕ ≤ e M . Since M is dual Baer, Imϕ ≤ ⊕ M . Hence, Imϕ = M . Therefore, according to ( [14], Lemma 3.14), M is continuous.  Proof. Suppose M is auto-invariant and c-retractable. Since M is dual Baer, we infer from Proposition 6(1) that M = ⊕ i∈I M i is a direct sum of extending modules. Thus, by Corollary 15 in [13], M is quasi-injective. The converse implication is clear.
Recall that an R-module M is called C 4 if, whenever A and B are submodules of M with M = A ⊕ B and f : A −→ B is an homomorphism with Kerf ≤ ⊕ A, we have Imf ≤ ⊕ B.
Proposition 9. If every 2-generated R-module is a C 4 -module, then every dual Baer c-retractable R-module is semisimple.

Proof.
Let M be any dual Baer c-retractable R-module. Thus, as in the proof of Theorem 4, M = ⊕ i∈I M i where each M i is uniform. Now, we have to show that each M i is semisimple. For any 0 = m ∈ E(M i ), let 0 = N ≤ mR and take 0 = n ∈ N . By our assumption, mR ⊕ nR is a C 4 -module. Consider the inclusion map i : nR −→ mR. Thus i(nR) = nR ≤ ⊕ mR. Since mR is indecomposable, nR = mR, and hence N = mR. Thus, every cyclic submodule of mR is a direct summand. It follows that mR is semisimple.
Theorem 5. The following conditions are equivalentes for a ring R: (1) R is semisimple artinian.
(2) ⇒ (1) Let I be a right ideal of R. Clearly, I ⊕R is c-retractable, and hence a C 4 -module by (2). Consider the inclusion map i : I −→ R. Therefore, i(I) = I ≤ ⊕ R. Hence, R R is semisimple. Thus, R is semisimple artinian. Consequently, S ≤ ⊕ F , and hence S is projective. Therefore, R is semisimple.
Recall that an R-module is called Baer if, for all N ≤ M , L S (N ) = Se, with e 2 = e ∈ S. A module M is called K-nonsingular if, ∀ϕ ∈ End(M ), Kerϕ ≤ e M implies ϕ = 0.
Proposition 10. Let M be a K-nonsingular c-retractable R-module. Then S is right nonsingular.
Proposition 11. Let M be a c-retractable R-module such that S S is extending. Then M is K-nonsingular if and only if M is Baer.

Proof.
Suppose M is K-nonsingular. By Proposition 10, S is right nonsingular. Let N be a submodule of M . Thus, L S (N ) is a complement right ideal in S. Because S S is extending, then L S (N ) = S(1 − e) for some e = e 2 ∈ S, and hence M is Baer. The converse implication follows from ( [18], Lemma 2.15).
Recall that a module is locally noetherian if any of its finitely generated submodules is noetherian. An R-module M is said to be homo-related to an R-module L if there are α : M −→ L and β : L −→ M such that βα = 0. Theorem 6. Let M be a locally noetherian c-retractable R-module. Then M is homorelated to a direct sum ⊕ i∈I U i of uniform submodules of M .   Proof. Suppose M has the stated condition. Let 0 = N ≤ M . Since U dim(N ) < ∞, N contains a uniform submodule U . After replacing U by an essential closure, we may assume that U is a complement submodule of M . By our assumption, there is a nonzero homomorphism M −→ U . Therefore, M is retractable.