\titleofpresentation{Modules whose Maximal Submodules are Supplements}

\presenter{Dilek Pusat-Y\i lmaz}(Joint work with Engin
B\"{u}y\"{u}ka\c{s}\i k)

Let $R$ be a ring and $M$ be a unital left $R$-module. $M$ is called
an $MS-module$  if every maximal submodule of $M$ is a supplement in
$M$, and $M$ is called an $MD-module$ if every maximal submodule of
$M$ is a direct summand of $M$.

In this talk some properties and some characterizations of
MS-modules will be given.
\begin{proposition}
Let $M=\sum_{i\in I} M_{i}$ for some index set $I$. Suppose $M_{i}$
is an $MS$-module for every $i\in I$. Then $M$ is an $MS$-module.
\end{proposition}

\begin{theorem}
Let $R$ be a noetherian ring and $M$ an $R$-module. $M$ is an
MS-module if and only if $M/D(M)$ has no maximal submodules where
\linebreak $D(M)=\sum \{Rm : m \in \Lambda\}$ and

\noindent $\Lambda =\{m \in M: Rm \cap P \ll P\,\, \forall
P\leqslant M\,\, \text{such that}\,\, Rm \nsubseteq P\}$.
\end{theorem}

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\begin{ourbiblio}{99}  %  Delete this environment if you
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\bibitem{pusat-yilmazAB} Alizade, R., B\"{u}y\"{u}ka\c{s}{\i}k, E. (2003).
Cofinitely weak supplemented modules \emph{Comm. Algebra}.
31(11):5377-5390.

\bibitem{pusat-yilmazAnderson} F.~W. Anderson, K.~R. Fuller, Rings and
Categories of Modules, (Springer, New York, 1992).

\bibitem{pusat-yilmazWisbauer} R. Wisbauer, Foundations of Modules and
Rings, (Gordon and Breach, 1991).

\bibitem{pusat-yilmazZoschinger:dedekind} H. Z\"{o}schinger, \emph{Komplementierte moduln \"{u}ber
Dedekindringen}, Journal of Algebra. 29 (1974), 42-56.


\end{ourbiblio}

\presenteraddress{Izmir Institute of Technology}  % The name of a university is fine.

\email{dilekyilmaz@iyte.edu.tr}


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