# Nebiyev C.

### $T$-radical and strongly $T$-radical supplemented modules

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1191-1196

We define (strongly) t-radical supplemented modules and investigate some properties of these modules. These modules lie between strongly radical supplemented and strongly $\oplus$ -radical supplemented modules. We also study the relationship between these modules and present examples separating strongly $t$-radical supplemented modules, supplemented modules, and strongly $\oplus$-radical supplemented modules.

### t-Generalized Supplemented Modules

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1491-1497

In the present paper, $t$-generalized supplemented modules are defined starting from the generalized ⨁-supplemented modules. In addition, we present examples separating the $t$-generalized supplemented modules, supplemented modules, and generalized ⨁-supplemented modules and also show the equality of these modules for projective and finitely generated modules. Moreover, we define cofinitely $t$-generalized supplemented modules and give the characterization of these modules. Furthermore, for any ring $R$, we show that any finite direct sum of $t$-generalized supplemented $R$-modules is $t$-generalized supplemented and that any direct sum of cofinitely $t$-generalized supplemented $R$-modules is a cofinitely $t$-generalized supplemented module.

### $G$-Supplemented Modules

Koşar B., Nebiyev C., Sökmez N.

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 861–864

Following the concept of generalized small submodule, we define $g$ -supplemented modules and characterize some fundamental properties of these modules. Moreover, the generalized radical of a module is defined and the relationship between the generalized radical and the radical of a module is investigated. Finally, the definition of amply $g$ -supplemented modules is given with some basic properties of these modules.

### On Supplement Submodules

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 961–966

We investigate some properties of supplement submodules. Some relations between lying-above and weak supplement submodules are also studied. Let *V* be a supplement of a submodule *U* in *M*. Then it is possible to define a bijective map between the maximal submodules of *V* and the maximal submodules of *M* that contain *U*. Let *M* be an *R*-module, *U* ≤ *M*, let *V* be a weak supplement of *U*, and let *K* ≤ *V*. In this case, *K* is a weak supplement of *U* if and only if *V* lies above *K* in *M*. We prove that an *R*-module *M* is amply supplemented if and only if every submodule of *M* lies above a supplement in *M*. We also prove that *M* is semisimple if and only if every submodule of *M* is a supplement in *M*.

### On strongly $\oplus$-supplemented modules

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 662-667

In this work, strongly $\oplus$-supplemented and strongly cofinitely $\oplus$-supplemented modules are defined and some properties of strongly $\oplus$-supplemented and strongly cofinitely $\oplus$-supplemented modules are investigated. Let $R$ be a ring. Then every $R$-module is strongly $\oplus$-supplemented if and only if R is perfect. Finite direct sum of $\oplus$-supplemented modules is $\oplus$-supplemented. But this is not true for strongly $\oplus$-supplemented modules. Any direct sum of cofinitely $\oplus$-supplemented modules is cofinitely $\oplus$-supplemented but this is not true for strongly cofinitely $\oplus$-supplemented modules. We also prove that a supplemented module is strongly $\oplus$-supplemented if and only if every supplement submodule lies above a direct summand.