2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64359We present short and elementary proofs of two theorems of Huckaba and Marley, while generalizing them at the same time to the case of a module. The theorems concern a characterization of the depth of the associated graded ring of a Cohen-Macaulay module, with respect to a Hilbert filtration, in terms of the Hilbert coefficient e_1. As an application, we derive bounds on the higher Hilbert coefficient e_i in terms of e_0.7 pages; This is a slightly modified version of the previously uploaded fileCommutative Algebra13H10, 13H15text