Let M be a right R-module. We call M Rad-H-supplemented iffor each Y M there exists a direct summand D of M such that(Y + D)/D (Rad(M) + D)/D and (Y + D)/Y (Rad(M) + Y )/Y .It is shown that:(1) Let M = M1M2, where M1 is a fully invariant submodule of M.If M is Rad-H-supplemented, thenM1 andM2 are Rad-H-supplemented.(2) Let M = M1 M2 be a duo module and Rad--supplemented. IfM1 is radical M2-...

a module $m$ is called $emph{h}$-cofinitely supplemented if for every cofinite submodule $e$ (i.e. $m/e$ is finitely generated) of $m$ there exists a direct summand $d$ of $m$ such that $m = e + x$ holds if and only if $m = d + x$, for every submodule $x$ of $m$. in this paper we study factors, direct summands and direct sums of $emph{h}$-cofinitely supplemented modules. let $m$ be an $emph{h}$...

In this paper we provide various properties of Rad-⊕-supplemented modules. In particular, we prove that a projective module M is Rad⊕-supplemented if and only if M is ⊕-supplemented, and then we show that a commutative ring R is an artinian serial ring if and only if every left R-module is Rad-⊕-supplemented. Moreover, every left R-module has the property (P ∗) if and only if R is an artinian s...

In this work, we introduce $H^*$-condition on the set of submodules of a module. Let $M$ be a module. We say $M$ satisfies $H^*$ provided that for every submodule $N$ of $M$, there is a direct summand$D$ of $M$ such that $(N+D)/N$ and $(N+D)/D$ are cosingular. We show that over a right perfect right $GV$-ring,a homomorphic image of a $H^*$ duo module satisfies $H^*$.

A module $M$ is called $emph{H}$-cofinitely supplemented if for every cofinite submodule $E$ (i.e. $M/E$ is finitely generated) of $M$ there exists a direct summand $D$ of $M$ such that $M = E + X$ holds if and only if $M = D + X$, for every submodule $X$ of $M$. In this paper we study factors, direct summands and direct sums of $emph{H}$-cofinitely supplemented modules. Let $M$ be an $emph{H}...

Lifting modules and their various generalizations as some main concepts in module theory have been studied and investigated extensively in recent decades. Some authors tried to present some homological aspects of lifting modules and -supplemented modules. In this work, we shall present a homological approach to -supplemented modules via fully invariant submodules. Lifting modules and H-suppleme...

In this paper, we introduce the concept of (amply) cofinitely ss-supplemented modules as a proper generalization modules, and provide various properties these modules. particular, prove that arbitrary sum is ss-supplemented. Moreover, show ring R semiperfect Rad(R)⊆Soc(RR) if only every left R-module