Transitivity for Weak and Strong Gröbner Bases

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) Gröbner basis in R[x] graded by \Gamma \Theta Z n if and only if F is a weak Gröbner basis in R[x] graded by f0g \Theta