MMN-1424

\begin{abstract} We say that over an arbitrary ring a module $M$ has \emph{the property} $(WE)$ (respectively, $(WEE)$) if $M$ has a weak supplement (respectively, ample weak supplements) in every extension. In this paper, we provide various properties of modules with these properties. We show that a module $M$ has the property $(WEE)$ iff every submodule of $M$ has the property $(WE)$. A ring $R$ is left perfect iff every left $R$-module has the property $(WE)$ iff every left $R$-module has the property $(WEE)$. A ring $R$ is semilocal iff every left $R$-module has a weak supplement in every extension with small radical. We also study modules that have a weak supplement(respectively, ample weak supplements) in every coatomic extension, namely \emph{the property} $(WE^*)$(respectively, $(WEE^*)$). \end{abstract}

Vol. 17 (2016), No. 1, pp. 471-481

DOI: 10.18514/MMN.2016.1424

Download: MMN-1424