2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65096We present a constructive description of minimal reductions with a given reduction number. This description has interesting consequences on the minimal reduction number, the big reduction number, and the core of an ideal. In particular, it helps solve a conjecture of Vasconcelos on the relationship between reduction numbers and initial ideals.15 pagesCommutative AlgebraAlgebraic Geometry13A02 (Primary) 13P10 (Secondary)text