# Pancar A.

### 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*.

### Generalizations of $\oplus$-supplemented modules

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 555-564

We introduce $\oplus$-radical supplemented modules and strongly $\oplus$-radical supplemented modules (briefly, $srs^{\oplus}$-modules) as proper generalizations of $\oplus$-supplemented modules. We prove that (1) a semilocal ring $R$ is left perfect if and only if every left $R$-module is an $\oplus$-radical supplemented module; (2) a commutative ring $R$ is an Artinian principal ideal ring if and only if every left $R$-module is a $srs^{\oplus}$-module; (3) over a local Dedekind domain, every $\oplus$-radical supplemented module is a $srs^{\oplus}$-module. Moreover, we completely determine the structure of these modules over local Dedekind domains.

### 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.

### On generalization of $⊕$-cofinitely supplemented modules

Ukr. Mat. Zh. - 2010. - 62, № 2. - pp. 183–189

We study the properties of ⊕-cofinitely radical supplemented modules, or, briefly, $cgs^{⊕}$-modules. It is shown that a module with summand sum property (SSP) is $cgs^{⊕}$ if and only if $M/w \text{Loc}^{⊕} M$ ($w \text{Loc}^{⊕} M$ is the sum of all $w$-local direct summands of a module $M$) does not contain any maximal submodule, that every cofinite direct summand of a UC-extending $cgs^{⊕}$-module is $cgs^{⊕}$, and that, for any ring $R$, every free $R$-module is $cgs^{⊕}$ if and only if $R$ is semiperfect.