 
Summary: TRANSITIVITY FOR WEAK AND STRONG GR š
OBNER BASES
W. W. ADAMS, A. BOYLE, AND P. LOUSTAUNAU
Abstract. Let R be a Noetherian integral domain which is graded by an
ordered group \Gamma 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) Gršobner
basis in R[x] graded by \Gamma \Theta Z n if and only if F is a weak Gršobner basis in
R[x] graded by f0g \Theta Z n and certain subsets of the set of leading coefficients
of the elements of F form weak (respectively strong) Gršobner bases in R. It
is further shown that any \Gammagraded ring R for which every ideal has a strong
Gršobner basis is isomorphic to k[x 1 ; : : : ; xn ], where k is a PID.
1. Introduction
Let y and x be sets of variables, each with a term order and an elimination order
between them. Let k denote a Noetherian commutative ring. In several places in
the literature (e.g. Adams and Boyle (1992), Bayer and Stillman (1987), Gianni,
Trager, and Zacharias (1988), Spear (1977), Shtokhamer (1988)), the problem of
lifting Gršobner bases from the ring k[y] to the ring k[y; x] has been examined.
This entails understanding the difference between a Gršobner basis in (k[y])[x] and
a Gršobner basis in k[y; x]. We refer to this as the transitivity question. This
problem was examined in Adams and Boyle (1992) mainly for the case when x = x
