Transitivity for Weak and Strong Gröbner Bases

Let R be a Noetherian integral domain which is graded by an ordered group Γ and let X be a set of n variables with a term order. It is shown that a finite subset F of R[X] is a weak (respectively strong) Gröbner basis in R[X] graded by Γ × Zn if and only if F is a weak Gröbner basis in R[X] graded by {0} × Zn and certain subsets of the set of leading coefficients of the elements of F form weak (respectively strong) Gröbner bases in R: It is further shown that any Γ-graded ring R for which every ideal has a strong Gröbner basis is isomorphic to k [x1,…,xn], where k is a PID.