on h-cofinitely supplemented modules

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 $emph{h}$-cofinitely supplemented module and let $n leq m$ be a submodule. suppose that for every direct summand $k$ of $m$, $(n + k)/n$ lies above a direct summand of $m/n$. then $m/n$ is $emph{h}$-cofinitely supplemented. let $m$ be an $emph{h}$-cofinitely supplemented module. let $n$ be a direct summand of $m$. suppose that for every direct summand $k$ of $m$ with $m=n+k$, $ncap k$ is also a direct summand of $m$. then $n$ is $emph{h}$-cofinitely supplemented. let $m = m_{1} oplus m_{2}$. if $m_{1}$ is radical $m_{2}$-projective (or $m_{2}$ is radical $m_{1}$-projective) and $m_{1}$ and $m_{2}$ are $emph{h}$-cofinitely supplemented, then $m$ is $emph{h}$-cofinitely supplemented