A Study of t-g-Radical Supplemented Modules

This study investigates a specific class of modules in terms of supplemented module theory, defined as $t$-$g$-radical supplemented modules. An $R$-module $M$ is called $t$-$g$-radical supplemented if every submodule of $M$ possesses a $g$-radical supplement that is also a $t$-summand of $M$. This research aims to provide a comprehensive structural analysis of these modules and establish several characterization theorems regarding their algebraic properties.

First, we establish the relationship between $t$-$g$-radical supplemented modules and classical supplemented modules. It is proved that if $M$ is a $t$-$g$-radical supplemented module such that $Rad_g(M)$ is a small submodule of $M$, then $M$ is necessarily a supplemented module. For finitely generated $R$-modules, we show that being $t$-$g$-radical supplemented implies being $t$-$g$-supplemented. One of the central results of this paper is the behavior of these modules under sums. We prove that a finite sum of $t$-$g$-radical supplemented modules remains $t$-$g$-radical supplemented. Furthermore, the preservation of this property under factor modules and homomorphic images is examined. Specifically, it is shown that for a distributive $t$-$g$-radical supplemented module $M$, every quotient module and every homomorphic image of $M$ inherits the $t$-$g$-radical supplemented property. Finally, we discuss the conditions under which every $t$-summand of $M$ is $t$-$g$-radical supplemented, particularly focusing on modules satisfying the $(D3)$ property or the Summand Sum Property (SSP).