  Let $R$ be a commutative ring. We investigate $R$-modules which can be
written as \emph{finite} sums of {\it {second}} $R$-submodules (we call them
\emph{second representable}). We provide sufficient conditions for an
$R$-module $M$ to be have a (minimal) second presentation, in particular within
the class of lifting modules. Moreover, we investigate the class of
(\emph{main}) \emph{second attached prime ideals} related to a module with such
a presentation.
