Abstract
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 Z2-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 C4-module are determined.
Highlights
Throughout all rings are associative with identity and all modules are unitary right module
With the help of c-retractability, we investigated when the notions of K-nonsingularity and Baer modules are equivalent
An R-module M is called continuous if it is extending and satisfies the following condition: (C2) Every submodule of M that is isomorphic to a direct summand M is itself a direct summand of M . 3 An R-module M is called quasi-continuous if it is extending and satisfies the following condition: (C3) If N and K are direct summands of M with N ∩ K = 0, N ⊕ K is a direct summand of M
Summary
Throughout all rings are associative with identity and all modules are unitary right module. Following [19], we say that an R-module M is retractable if HomR(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. 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. We are going to show among others, the following results: (1) Every projective module over a right SI-ring is c-retractable. The notations N ≤ M , N ≤e M and N ≤⊕ M mean that N is a submodule of M , an essential submodule and a direct summand of M , respectively
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