Gröbner bases and products of coefficient rings

Suppose that A is a finite direct product of commutative rings. We show from first principles that a Gröbner basis for an ideal of A[x1,…,xn] can be easily obtained by ‘joining’ Gröbner bases of the projected ideals with coefficients in the factors of A (which can themselves be obtained in parallel)...

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Bibliographic Details
Published in:Bulletin of the Australian Mathematical Society 2002-02, Vol.65 (1), p.145-152
Main Authors: Norton, Graham H., Sӑlӑgean, Ana
Format: Article
Language:English
Subjects:
Coefficients
First principles
Rings (mathematics)
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Summary:Suppose that A is a finite direct product of commutative rings. We show from first principles that a Gröbner basis for an ideal of A[x1,…,xn] can be easily obtained by ‘joining’ Gröbner bases of the projected ideals with coefficients in the factors of A (which can themselves be obtained in parallel). Similarly for strong Gröbner bases. This gives an elementary method of constructing a (strong) Gröbner basis when the Chinese Remainder Theorem applies to the coefficient ring and we know how to compute (strong) Gröbner bases in each factor.
ISSN:0004-9727
1755-1633
DOI:10.1017/S0004972700020165