2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/228382We present basic properties of Gröbner bases of submodules of a free module of finite rank over a polynomial ring $R$ with coefficients in a graded truncated discrete valuations ring $A$. As an application, we give a criterion for a finitely generated $R$-module to be flat over $A$. Its non-graded version is also given.Commutative AlgebraNumber Theory13P10; 13C11; 11S15text