Degenerations over $(A_{\mathrm{\infty }})$-singularities and construction of degenerations over commutative rings

We give a necessary condition of degeneration via matrix representations, and consider degenerations of indecomposable Cohen-Macaulay modules over hypersurface singularities of type ($A_{\mathrm{\infty }}$). We also provide a method to construct degenerations of finitely generated modules over commutative rings.