\titleofpresentation{On Cofinitely Weak Supplemented Modules}

\presenter{Y{\i}lmaz Mehmet Demirci}  %  Name just one speaker, please; collaborators
            %  can be credited in the body of the abstract.

%  [Body of abstract goes here.]

An \emph{R}-module \emph{M} is called weakly supplemented, if every
submodule \emph{N} of \emph{M} has a \emph{weak supplement}, i.e. a
submodule \emph{K} satisfying $N+K=M$ and $N\cap K\ll M$. Semisimple
modules and artinian modules are weakly supplemented. A submodule
\emph{N} of a module \emph{M} is said to be \emph{cofinite} if $M/N$
is finitely generated. \emph{M} is called a \emph{cofinitely weak
supplemented} (briefly \emph{cws}) module if every cofinite
submodule of \emph{M} has a weak supplement. An \emph{R}-module
\emph{M} is called \emph{locally noetherian} if every finitely
generated submodule of \emph{M} is noetherian.

\begin{theorem}
Let $M$ be a locally noetherian module with $\operatorname{Rad} M
\ll M$. The following statements are equivalent.
\begin{enumerate}

\item $M$ is weakly supplemented.
\item Every cyclic submodule of $M$ has (is) a weak supplement.
\item Every finitely generated submodule of $M /\operatorname{Rad} M$ is a
direct summand.
\end{enumerate}
\end{theorem}

\begin{theorem}
For a locally noetherian module $M$, the following are equivalent.

\begin{enumerate}
\item $M$ is a \emph{cws}-module.
\item $M/X$ is a \emph{cws}-module for
some $X \subseteq \operatorname{Rad} M.$
\item $M/\operatorname{Rad} M$ is a
\emph{cws}-module.
\end{enumerate}
\end{theorem}
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{demirciABS} R. Alizade, G. Bilhan, P.F. Smith, ``Modules whose maximal
submodules have supplements", \emph{Comm.  Algebra},  29 (2001), pp.
2389-2405.
\bibitem{demirciAB} R. Alizade, E. B\"{u}y\"{u}ka\c{s}{\i}k,
 ``Cofinitely weak supplemented modules'', \emph{Comm. Algebra}, 31 (2003), pp. 5377-5390.

\bibitem{demirciAnderson} F.~W. Anderson, and K.~R. Fuller, \emph{Rings and
Categories of Modules}. New York: Springer-Verlag, 1992.

\bibitem{demirciLam} T.Y. Lam,  \emph{A First Course in noncommutative
rings}. New York:Springer-Verlag, 2001.
\bibitem{demirciLomp} C. Lomp, ``On semilocal modules and rings'', \emph{Comm.
Algebra}, 27  (1999), pp. 1921-1935.
\bibitem{demirciMatlis} E. Matlis, ``Decomposable modules'', \emph{Trans.
Am. Math. Soc.}, 125 (1966), pp. 147-179.
\bibitem{demirciSharp} D.W. Sharpe, P. Vamos, \emph{Injective Modules}. Cambridge
University Press, 1972.
\bibitem{demirciWisbauer} R. Wisbauer, \textit{Foundations of Modules and
Rings}. London: Gordon and Breach Science Publishers, 1991.
\bibitem{demirciZoschinger Inv} H. Z\"{o}schinger, ``Invarianten wesentlicher
\"{U}berdeckungen'', \emph{Math. Annalen}, 237 (1978), pp. 193-202.


\end{ourbiblio}

\presenteraddress{\.{I}zmir Institute of Technology, Department of
Mathematics, G\"{u}lbah\c{c}e K\"{o}y\"{u}, \.{I}zmir}  % The name
				% of a university is fine. 

\email{yilmazdemirci@iyte.edu.tr}

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

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