2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/94859We interpret the Artin-Rees lemma and the Izumi theorem in term of Artin function and we obtain a stable version of the Artin-Rees lemma. We present different applications of these interpretations. First we show that the Artin function of $X_1X_2-X_3X_4$, as a polynomial in the ring of power series in more than three variables, is not bounded by an affine function. Then we prove that the Artin functions of a class of polynomials are bounded by affine functions and we use this to compute approximated integral closures of ideals.to appear in J. of Algebra; a lot of misprints changed and some proofs are improved following referee's commentsCommutative AlgebraAlgebraic Geometrytext