  Let 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.
