Transitivity for Weak and Strong Gröbner Bases

Abstract 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) Grobner basis in R [X] graded by Γ × Z n if and only if F is a weak Grobner basis in R [X] graded by {0} × Z n and certain subsets of the set of leading coefficients of the elements of F form weak (respectively strong) Grobner bases in R: It is further shown that any Γ-graded ring R for which every ideal has a strong Grobner basis is isomorphic to k [ x 1 ,…, x n ], where k is a PID.