Based on an explicit description of the idealization of a graded submodule of a graded free module, we examine the behaviour of Gröbner bases and minimal homogeneous systems of generators under this process. Then we show how one can idealize a homogeneous presentation. Using this theory, we present a unified description of several strategies for computing minimal homogeneous presentations and minimal graded free resolutions, in particular of the vertical and horizontal strategies. We obtain at a simple and compact algorithm for computing minimal graded free resolutions degree‐by‐degree which lends itself well to further optimizations.