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Vietnam Journal of Mathematics 35:2(2007) 215-222  H-Cofinitely Supplemented Modules Muhammet Tamer Kosan Abstract.  Let $M$ be a right $R$-module. We call $M$ {\it $H$-cofinitely supplemented} if for very cofinite submodule $A$ of $M$ (i.e. the factor module $M/A$ is finitely generated) there exists a direct summand $D$ of $M$ such that $M=A+X$ holds if and only if $M=D+X$. It is shown that in this paper: $(1)$ Let $M$ be an $H$-cofinitely supplemented Duo module. Then every direct summand of $M$ is $H$-cofinitely supplemented module. $(2)$ Let $M=M_1\oplus M_2$ be a Duo module. If $M_1$ and $M_2$ are $H$-cofinitely supplemented modules, then $M$ is $H$-cofinitely supplemented. $(3)$ Assume $Rad(M)<< M$. Then $M$ is $H$-cofinitely supplemented if and only if every cofinite submodule of $M/Rad(M)$ is direct summand and each cofinite direct summand of $M/Rad(M)$ lifts to a direct summand of $M$. In addition, let $M$ be a duo right $R$-module. $M$ is $H$-cofinitely supplemented if and only if every maximal submodule of $M$ has an $H$-supplement in $M$.   2000 Mathematics Subject Classification: 16D99. Keywords: Duo modules, $H$-supplemented modules, $H$-cofinitely supplemented modules.

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