Generalized ⊕-Radical Supplemented Modules

Çalışıcı and Türkmen called a module M generalized ⊕-supplemented if every submodule has a generalized supplement that is a direct summand of M. Motivated by this, it is natural to introduce another notion that we called generalized ⊕-radical supplemented modules as a proper generalization of generalized ⊕-supplemented modules. In this paper, we obtain various properties of generalized ⊕-radical supplemented modules. We show that the class of generalized ⊕-radical supplemented modules is closed under finite direct sums.We attain that over a Dedekind domain amoduleM is generalized ⊕-radical supplemented if and only ifM/P(M) is generalized ⊕-radical supplemented. We completely determine the structure of these modules over left V-rings. Moreover, we characterize semiperfect rings via generalized ⊕-radical supplemented modules.


Introduction
Throughout the whole text, all rings are to be associative; unit and all modules are left unitary.We specially mention [1][2][3][4] among books concerning the structures of modules and rings.We shall write  ≤  ( ≪ ) if  is a submodule of  (small in ).By Rad(), we denote the radical of .Let ,  ≤ . is called a supplement of  in  if it is minimal with respect to  =  + . is a supplement of  in  if and only if  =  +  and  ∩  ≪  [4].A module  is called supplemented if every submodule of  has a supplement, and it is called ⊕supplemented if every submodule of  has a supplement that is a direct summand of  [5].Clearly every ⊕-supplemented module is supplemented.In [6], Zöschinger introduced a notion of modules whose radical has supplements called radical supplemented.Xue defined generalized supplemented modules as another generalization of supplemented modules [7].Let  be any submodule of .If there exists a submodule  of  such that  =  +  and  ∩  ⊆ Rad(),  is called a generalized supplement of  in . is called generalized supplemented if every submodule of  has a generalized supplement in .Also C ¸alıs ¸ıcı and Türkmen called a module  generalized ⊕-supplemented if every submodule has a generalized supplement that is a direct summand of  as a generalization of ⊕-supplemented modules [8].So it is natural to introduce another notion that we called generalized ⊕-radical supplemented modules.A module  is called generalized ⊕-radical supplemented if every submodule containing radical has a generalized supplement that is a direct summand of .
In this paper we obtain various properties of generalized ⊕-radical supplemented modules as a proper generalization of generalized ⊕-supplemented modules.We prove the following indications.
(i) Every generalized ⊕-radical supplemented module has a radical direct summand.
(iii) The class of generalized ⊕-radical supplemented modules is closed under finite direct sums.
(iv) If  is a generalized ⊕-radical supplemented module, then / is a generalized ⊕-radical supplemented module for every fully invariant submodule  of .

ISRN Algebra
If  is a duo module, then  is generalized ⊕-radical supplemented. (

Generalized ⊕-Radical Supplemented Modules
Definition 1.A module  is called generalized ⊕-radical supplemented if every submodule containing radical has a generalized supplement that is a direct summand of .
Recall that a module  is called radical if  has no maximal submodules; that is, Rad() = .For a module , () will indicate the sum of all radical submodules of .If () = 0,  is called reduced.Note that () is the largest radical submodule of  [4].Now we have the following simple fact, which lays a key role in our study.Recall from [4] that a submodule  of an -module  is called fully invariant if () is contained in  for every endomorphism  of .A module  is called duo, if every submodule of  is fully invariant [9].Theorem 5.The following statements hold over a ring .
Corollary 18.Let  be a local commutative ring.Suppose that every submodule of (/ Rad ()) is generalized ⊕-radical supplemented, where (/ Rad ()) is the injective hull of the simple module / Rad ().Then  is a uniserial ring.