# On H-cofinitely supplemented modules

Document Type : Research Paper

Authors

1 University of Mazandaran, Iran

2 University of Tetouan

3 Univeristy of Mazandaran, Iran

Abstract

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

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### History

• Receive Date: 16 August 2011
• Revise Date: 29 March 2012
• Accept Date: 12 April 2012