t-g-radical supplemented modules

Abstract

We introduce and study the class of $t$-$g$-radical supplemented modules, which unifies two independent generalizations of the classical supplemented module condition: $g$-radical supplements and $t$-sum terms. A module $M$ is $t$-$g$-radical supplemented if every submodule $N \leq M$ has a $g$-radical supplement that is simultaneously a $t$-sum term of $M$. We establish closure properties under $t$-sums, quotient modules, and homomorphic images, prove inheritance by $t$-sum terms, and classify the standard module classes (simple, semisimple, local, hollow, and Prüfer groups) within this framework. A key observation is that the class strictly contains the class of supplemented modules: both $\mathbb{Q}$ and $\mathbb{Z}_{p^{\infty}}$ are $t$-$g$-radical supplemented but not supplemented.