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Transitivity for Weak and Strong Gröbner Bases

✍ Scribed by W.W. Adams; A. Boyle; P. Loustaunau

Publisher
Elsevier Science
Year
1993
Tongue
English
Weight
563 KB
Volume
15
Category
Article
ISSN
0747-7171

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✦ Synopsis

Let (R) be a Noetherian integral domain which is graded by an ordered group (\Gamma) and let (\mathbf{x}) be a set of (n) variables with a term order. It is shown that a finite subset (F) of (R[\mathbf{x}]) is a weak (respectively strong) Gröbner basis in (R[\mathbf{x}]) graded by (\Gamma \times \mathbf{Z}^{n}) if and only if (F) is a weak Gröbner basis in (R[\mathbf{x}]) graded by ({0} \times \mathbf{Z}^{n}) and certain subsets of the set of leading coefficients of the elements of (F) form weak (respectively strong) Gröbner bases in (R) : It is further shown that any (\Gamma)-graded ring (R) for which every ideal has a strong Gröbner basis is isomorphic to (k\left[x_{1}, \ldots, x_{n}\right]), where (k) is a PID.

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