2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/60817We present a proof of embedded desingularization for closed subschemes which does not make use of Hilbert-Samuel function and avoids Hironaka's notion of normal flatness. This proof, already sketched in [A course on constructive desingularization and equivariance. In {\em Resolution of singularities (Obergurgl, 1997)}, vol. 181 {\em Progr. Math.}, Birkhäuser, 2000.] page 224, is done by showing that desingularization of a closed subscheme $X$, in a smooth sheme W, is achieved by taking an algorithmic principalization for the ideal $I(X)$, associated to the embedded scheme $X$.In accordance to the suggestions of referee: Title has changed and the structure of the paper is different. Proof of main theorem is clarified. Latex document, 11pagesAlgebraic Geometry14E15; 32S45text