2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78749For two ideals $I$ and $J$ of a noetherian ring, we characterize, in terms of the vanishing of Tor modules, when the associated graded ring of the sum $I+J$ is isomorphic to the tensor product of the associated graded ring of $I$ and the associated graded ring of $J$. It is shown that the relation type of the tensor product of two standard algebras is bounded above by the maximum of the relation type of each algebra. As a consequence, we deduce a uniform bound for the relation type of maximal ideals of an excellent ring and a classical result of Duncan and O'Carroll on the strong uniform Artin-Rees property.12 pages, LatexCommutative AlgebraAlgebraic Geometry13A30 (Primary) 13D02 (Secondary)text