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Copyright of Adiyaman University Journal of Science & Technology / Adıyaman Üniversitesi Fen Bilimleri Dergisi is the property of Adiyaman University, Institute of Science / Adiyaman Universitesi Fen Bilimleri Enstitusu and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let R be a ring and M be a left R-module. M is called cofinitely weak δ-supplemented (or briefly δ- CWS-module) if every cofinite submodule of M has a weak δ-supplement in M. In this paper, we give various properties of this kind of modules. It is shown that a module M is δ-CWS-module if and only if every maximal submodule has a weak δ-supplement in M. The class of cofinitely weak δ-supplemented modules are closed under taking homomorphic images, arbitrary sums and short exact sequences. Also we give some conditions equivalent to being a δ-CWS-module for a δ-coatomic module. [ABSTRACT FROM AUTHOR]

We study modules with the properties (δ - TWEE) and (δ - TWEE) which are adopted Zöschinger's modules with the properties E and E. We call a module(δ - TWEE) module if M has a weak M-supplement in every torsion extension. Similarly if M has ample weak δ-supplements in every torsion extension then M is called (δ - TWEE) module. We obtain various properties of these modules. We will show that (1) Every direct summand of a (δ - TWEE) module is a (δ - TWEE) module. (2) A module M has the property(δ - TWEE) if every submodule of M has the property(δ - TWEE). (3) Any factor module of a (δ - TWEE) module is a (δ - TWEE) module under a special condition. (4) Over a nonlocal ring, if every submodule of a module ½ is a (δ - TWEE) module, then it is cofinitely weak δ-supplemented. [ABSTRACT FROM AUTHOR]

In this paper, necessary and sufficient conditions for a quotient module are found to be a cofinitely weak Rad-supplemented module under which circumstances. Nevertheless, some relations are investigated between cofinitely Rad-supplemented modules and cofinitely weak Rad-supplemented modules. Lastly, we show that an arbitrary ring R is a left Noetherian V-ring if and only if every weak Rad-supplemented R-module is injective. [ABSTRACT FROM AUTHOR]

In this paper, we introduce strongly generalized (weakly) δ-supplemented modules. We call a module strongly generalized (weakly) δ-supplemented (brie y δ-SGS (δ-SWGS)) if every sub-module containing the δ-radical has a (weak) δ-supplement. The first part of this paper investigates various properties of δ-SGS modules. We prove that δ-SGS modules are closed under factor modules and finite sums.Using these modules, we show that a ring R is δ-semiperfect if and only if every left R-module is a δ-SGS module. The second part of this paper establishes some properties of δ-SWGS modules. [ABSTRACT FROM AUTHOR]

In this paper, we will define δ-small soft submodules and investigate the properties of them by using soft module theory. Moreover, we will introduce the notion of δ-radical of a soft module and will prove some basic properties of δ-radicals of a soft module. [ABSTRACT FROM AUTHOR]

In this paper, we will define δ−small soft submodules and investigate the properties of them by using soft module theory. Moreover, we will introduce the notion of δ−radical of a soft module and will prove some basic properties of δ−radicals of a soft module. [ABSTRACT FROM AUTHOR]