Grobner bases in orders of algebraic number fields

We prove that any order O of any algebraic number field K is a reduction ring. Rather than showing the axioms for a reduction ring hold, we start from scratch by well-ordering O, defining a division algorithm, and demonstrating how to use it in a Buchberger algorithm which computes a Grobner basis given a finite generating set for an ideal. It is shown that our theory of Grobner bases is equivalent to the ideal membership problem and in fact, a total of eight characterizations are given for a Grobner basis. Additional conclusions and questions for further investigation are revealed at the end of the paper. (C) 2002 Academic Press.