A generalization of reduction rings
Journal of Symbolic Computation
Gro¨bner bases and primary decomposition of modules
Journal of Symbolic Computation
A course in computational algebraic number theory
A course in computational algebraic number theory
Algorithmic properties of polynomial rings
Journal of Symbolic Computation
A critical-pair/completion algorithm for finitely generated ideals in rings
Proceedings of the Symposium "Rekursive Kombinatorik" on Logic and Machines: Decision Problems and Complexity
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
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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 Grbner basis given a finite generating set for an ideal. It is shown that our theory of Grbner bases is equivalent to the ideal membership problem and in fact, a total of eight characterizations are given for a Grbner basis. Additional conclusions and questions for further investigation are revealed at the end of the paper. Copyright 2002 Academic Press