2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72664Let R be a commutative ring with unity and a let A be a not necessarily commutative R-algebra which is free as an R-module. If I is an ideal in A, one can ask when A/I is also free as an R-module. We show that if A has an admissible system and I has a unital Grobner basis then A/I is free as an R-module. We prove a version of Buchberger's theorem over R and, as a corollary, we obtain a Grobner basis proof of the Poincare-Birkhoff-Witt Theorem over a commutative ground ring.10 pagesRings and Algebras16Z05; 13P10text