Generalized -Radical Supplemented Modules

？al？？？c？ and Türkmen called a module generalized -supplemented if every submodule has a generalized supplement that is a direct summand of . 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 a module is generalized -radical supplemented if and only if is generalized -radical supplemented. We completely determine the structure of these modules over left -rings. Moreover, we characterize semiperfect rings via generalized -radical supplemented modules. 1. Introduction Throughout the whole text, all rings are to be associative; unit and all modules are left unitary. We specially mention [1–4] among books concerning the structures of modules and rings. We shall write ( ) if is a submodule of (small in ). By , 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], Z？schinger 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 , is called a generalized supplement of in . is called generalized supplemented if every submodule of has a generalized supplement in . Also ？al？？？c？ 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.(ii) is a