By Martin Kreuzer

Bridges the present hole within the literature among idea and actual computation of Groebner bases and their functions. A accomplished consultant to either the idea and perform of computational commutative algebra, excellent to be used as a textbook for graduate or undergraduate scholars. comprises tutorials on many matters that complement the cloth.

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Extra resources for Computational Commutative Algebra 1

Example text

8. Let R be a factorial domain and f1 , . . , fm ∈ R \ {0} . a) The element lcm(f1 , . . , fm ) generates the ideal (f1 ) ∩ · · · ∩ (fm ). b) We have gcd(f1 , f2 ) = f1 f2 / lcm(f1 , f2 ). c) Suppose R is a principal ideal domain. Then gcd(f1 , . . , fm ) generates the ideal (f1 , . . , fm ). In particular, we have gcd(f1 , . . , fm ) = 1 if and only if there are elements g1 , . . , gm ∈ R such that g1 f1 +· · ·+gm fm = 1 . Proof. Since least common multiples were defined recursively, it suffices to αs 1 prove claim a) for m = 2 .

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X1 5 x1 • Dickson’s Lemma can be generalized to monomial modules as follows. 9. (Structure Theorem for Monomial Modules) Let M ⊆ P r be a monomial module. e. there are finitely many terms t1 , . . , ts ∈ Tn and numbers γ1 , .

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