\titleofpresentation{Radical supplemented modules}
\presenter{E. T\"{u}rkmen}

This is joint work with A. Pancar, and F.Y\"{u}zba\c{s}\i.
Let $R$ be a ring , M a  left  R - module  and  U a submodule
 of M. It  is  known  that  V  a  submodule  of  M  is called  a
supplement  of  U  in  M  if  U $cap $ V $<<$ V  with  U  +  V
 =  M.  We  define that  V  a  submodule  of  M  is called  a
Radical  supplement  of  U  in  M  if  U $cap $ V $\subseteq $ Rad
V  with  U  +  V  =  M.  In   this  paper   equality  of  the
radical supplement  is  shown  and   various   properties  of  both
 the radical supplement and  the submodule  of  M   has a radical 
supplement  in M  are  given.Similary, we  define  that  M  is 
called  radical  supplemented  if  every  submodule  U of  M  has 
a  radical  supplement in  M. Several properties of  the  radical
supplemented  mod{u}les  are  studied.

\begin{theorem}
Let  $M$  be  an  $R$ - module  and  $U,V\leq  M.$ $V$  is  a 
Rad  -  supplement  of  $U$  if  and  only  if   $U+V=M$  and  $%
<m>=Rm<<V $, for  each  $m\in  U\cap  V.$
\end{theorem}

\begin{theorem}
Let  $M$  be  an  $R$- modul  and  $U\leq  M$ . If  $U<<M$,  then  
$M$ is a Rad - supplement of $U$.
\end{theorem}

\begin{lemma}
Let  $M$  be  an  $R$- module  and  $V$  be a Rad - supplement  of $%
U $  in $M$.  If  $K<<M$,   then  $K\cap V\subset  RadV.$
\end{lemma}

\begin{theorem}
Let  $M$  be  an  $R$- module  and $V$  be a Rad - supplement  of $%
U $  in $M$.   If  $RadM<<M,  RadV=V\cap  RadM.$
\end{theorem}

\begin{theorem}
Let $ M$  be  an  $R$- module  and  $V$  be a Rad - supplement  of $%
U $  in $M$.   If $ L\subset  U$ ,  then $(V+L)/L$ is a  Rad -
supplement  of  $U/L$  in $M/L.$
\end{theorem}

\begin{theorem}
Let $M$ be an  $R$ - module  and  $M_{1},$ $M_{2}  \leq   M$.  If  $%
M_{1}$  is  Rad  supplemented  and  $M_{2}$  is  hollow , then $
M_{1} +M_{2\text{ \ }}$has  a Rad - supplement  in  $M$
\end{theorem}

\begin{theorem}
Let  $M$ be  an $ R$- module  and $ K<L<M.$If $ L$  is  local  and
 $L/K$  is a Rad - supplement  in $ M/K$,  then $ L$   is  a Rad -
supplement  in $M.$
\end{theorem}

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\presenteraddress{Ondokuz May\i s University,
Faculty of Sciences and Arts,
Department of Mathematics,
55139 Kurupelit, Samsun, TURKEY}

\email{ergulturkmen@hotmail.com.tr}

