ss-lifting modules and rings

  • Figen Eryilmaz, Ondokuz Mayis University, Department of Mathematics Education, Kurupelit, Atakum, 55139 Samsun, Turkey,


A module $M$ is called \emph{$ss$-lifting} if for every submodule $A$ of $M$, there is a decomposition $M=M_{1}\oplus M_{2\text{ }}$ such that $M_{1}\leq A$ and $A\cap M_{2}\subseteq Soc_{s}\left( M\right) $, where $Soc_{s}(M)=Soc(M)\cap Rad(M)$. In this paper, we provide the basic properties of $ss$-lifting modules. It is shown that: $(1)$ a module $M$ is $ss$-lifting iff it is amply $ss$-supplemented and its $ss$-supplement submodules are direct summand; $(2)$ for a ring $R$, $_{R}R$ is $ss$-lifting if and only if it is $ss$-supplemented iff it is semiperfect and its radical is semisimple; $(3)$ a ring $R$ is a left and right artinian serial ring and $Rad\left( R\right)\subseteq Soc\left( _{R}R\right)$ iff every left $R$-module is $ss$-lifting. We also study on factor modules of $ss$-lifting modules.

Vol. 22 (2021), No. 2, pp. 655-662
DOI: 10.18514/MMN.2021.3245

Download: MMN-3245