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\titleofpresentation{Extensions of weakly supplemented modules}

\presenter{Rafail Alizade}  %  Name just one speaker, please; collaborators
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%  [Body of abstract goes here.]
Throughout, $R$ is a ring with identity and modules are  unital left
$R$-modules. If $N + K = M$ and $N \cap K \ll M,$ then $K$ is called
a {\it weak supplement} of $N$ (see, \cite{Alizade2}). $M$ is a {\it
weakly supplemented} module if every submodule of $M$ has a weak
supplement. A submodule $N$ of $M$ is called \emph{coclosed} in $M$
if $N/K \ll M/K$ for some $K\subseteq M$ then $K=N.$

\begin{theorem}\label{extension} Let $0\rightarrow L \rightarrow M \rightarrow N \rightarrow
0$ be a short exact sequence. If $L$ and $N$ are weakly supplemented
and $L$ has a weak supplement in $M$ then $M$ is weakly
supplemented.

If $L$ is coclosed in $M$ then the converse holds, that is if $M$ is
weakly supple\-men\-ted then $L$ and $N$ are weakly supplemented.
\end{theorem}

The hypothesis that $L$ has a weak supplement can not be omitted. To
give an example we firstly prove the following lemmas.

\begin{lemma}\label{lemma:K is weakly supplemented } Let $R$ be a Dedekind domain and $K$ be the field of
quotients of $R$. Then $_{R}K$ is weakly supplemented.
\end{lemma}

\begin{lemma}\label{proposition:M over T is divisible} Let $R$
be a Dedekind domain and $\{\mathfrak{p}_{i}\}_{i \in I}$ be an
infinite collection of distinct maximal ideals of $R$. Let $M=\prod
\limits_{i\in I}(R/\mathfrak{p}_{i})$ be the direct product of the
simple $R$-modules $R/\mathfrak{p}_{i}$ and $T= T(M)$ be the torsion
submodule of $M$. Then $M/T$ is divisible, therefore $M/T\cong
K^{(J)}$ for some index set $J$ and $\operatorname{Rad} M=0$.
\end{lemma}


\begin{example} \label{Example}
Let $R$ and $M$ be as in Lemma \ref{proposition:M over T is
divisible} and  $T=\oplus_{i \in I}(R/\mathfrak{p}_{i})$ be the
torsion submodule of $M$. Note that $T$ is semisimple, so it is
weakly supplemented. Let $N$ be a submodule of $M$ such that $N/T
\cong K$. Then $N/T$ is weakly supplemented by Lemma \ref{lemma:K is
weakly supplemented }. Note that $\operatorname{Rad} N =0$ by Lemma
\ref{proposition:M over T is divisible} and  $N$ is not semisimple
because $N/T \cong K $ is not semisimple. Hence by Corollary 2.3 in
\cite{Alizade1}, $N$ is not weakly supplemented.
\end{example}

Moreover we have proved that a Dedekind domain $R$ is semilocal iff
every direct product of simple $R$-modules is weakly supplemented.

This is joint work with Engin B\"{u}y\"{u}ka\c{s}{\i}k, \.{I}zmir
Institute of Technology,\\
\url{enginbuyukasik@iyte.edu.tr}.

\begin{ourbiblio}{99}  %  Delete this environment if you
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\bibitem{Alizade1} C. Lomp, \emph{On semilocal modules and rings},
Communications in Algebra. 27:4 (1999), 1921-1935.
\bibitem{Alizade2} H. Z\"{o}schinger, \emph{Invarianten wesentlicher
\"{U}berdeckungen}, Math. Annalen. 237 (1978), 193-202.


\end{ourbiblio}

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

\email{rafailalizade@iyte.edu.tr}

\web{www.iyte.edu.tr/~rafailalizade}  % Those without homepages should delete this.

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