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On a generalization of $\oplus$-supplemented modules
Burcu Nisanci Turkmen,Bijan Davvaz 호남수학회 2019 호남수학학술지 Vol.41 No.3
We introduce \emph{FI-$\oplus$-supplemented} modules as a proper generalization of $\oplus$-supplemented modules. We prove that; $(1)$ every finite direct sum of FI-$\oplus$-supplemented $R$-modules is an FI-$\oplus$-supplemented $R$-module for any ring $R$ ; $(2)$ if every left $R$-module is FI-$\oplus$-supplemented over a semilocal ring $R$, then $R$ is left perfect; $(3)$ if $M$ is a finitely generated torsion-free uniform $R$-module over a commutative integrally closed domain such that every direct summand of $M$ is FI-$\oplus$-supplemented, then $M$ is a direct sum of cyclic modules.
ON A GENERALIZATION OF ⊕-SUPPLEMENTED MODULES
Turkmen, Burcu Nisanci,Davvaz, Bijan The Honam Mathematical Society 2019 호남수학학술지 Vol.41 No.3
We introduce FI-${\oplus}$-supplemented modules as a proper generalization of ${\oplus}$-supplemented modules. We prove that; (1) every finite direct sum of FI-${\oplus}$-supplemented R-modules is an FI-${\oplus}$-supplemented R-module for any ring R ; (2) if every left R-module is FI-${\oplus}$-supplemented over a semilocal ring R, then R is left perfect; (3) if M is a finitely generated torsion-free uniform R-module over a commutative integrally closed domain such that every direct summand of M is FI-${\oplus}$-supplemented, then M is a direct sum of cyclic modules.
F<sub>δ</sub>-SUPPLEMENTED MODULES
Turkmen, Burcu Nisanci,Eryilmaz, Figen The Honam Mathematical Society 2020 호남수학학술지 Vol.42 No.2
In this article, we define a (an amply) F<sub>δ</sub>-supplemented module in category of R-Mod. The general properties of F<sub>δ</sub>-supplemented modules are briefly discussed. Then, concentrating on the F<sub>δ</sub>-small submodule, we find the necessary and sufficient condition for F<sub>δ</sub>- supplemented modules. Also, we introduce ascending chain condition for F<sub>δ</sub>-small submodules of any module and establish a basic theorem for amply F<sub>δ</sub>-supplemented modules by using π-projectivity.
ON A GENERALIZATION OF □-SUPPLEMENTED MODULES
( Burcu Nisanci Turkmen ),( Bijan Davvaz ) 호남수학회 2019 호남수학학술지 Vol.41 No.3
We introduce FI-□-supplemented modules as a proper generalization of □-supplemented modules. We prove that; (1) every _nite direct sum of FI-□-supplemented R-modules is an FI-□-supplemented R-module for any ring R ; (2) if every left R-module is FI-□-supplemented over a semilocal ring R, then R is left perfect; (3) if M is a finitely generated torsion-free uniform R-module over a commutative integrally closed domain such that every direct summand of M is FI-□-supplemented, then M is a direct sum of cyclic modules.