2019
Том 71
№ 6

# Türkmen B. N.

Articles: 2
Article (English)

### A generalization of semiperfect modules

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 104-112

A module $M$ is called radical semiperfect, if $\frac MN$ has a projective cover whenever $\mathrm{R}\mathrm{a}\mathrm{d}(M) \subseteq N \subseteq M$. We study various properties of these modules. It is proved that every left $R$-module is radical semiperfect if and only if $R$ is left perfect. Moreover, radical lifting modules are defined as a generalization of lifting modules.

Article (English)

### 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.