  Let $R$ be a ring with unity, $\sigma$ an endomorphism of $R$ and $M_R$ a
right $R$-module. In this paper, we continue studding $\sigma$-rigid modules
that were introduced by Gunner et al. \cite{generalized/rigid}. We give some
results on $\sigma$-rigid modules and related concepts. Also, we study the
transfer of $\sigma$-rigidness from a module $M_R$ to its extensions, as
triangular matrix modules and polynomial modules.
