2019
Том 71
№ 4

# Nebiyev C.

Articles: 5
Article (English)

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

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.

Article (English)

### 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.

Brief Communications (English)

### $G$-Supplemented Modules

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.

Article (English)

### 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.

Article (English)

### 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.