# zbMATH — the first resource for mathematics

Computing the conductor of an integral extension. (English) Zbl 0755.13012
Summary: We describe algorithms to solve different problems in commutative algebra. These algorithms are linked by the fact that they are useful in considering integral extensions (integral closure, weak integral closure, etc.).
The first algorithm computes the conductor of an extension of rings. The algorithm is based on the computation of the inverse image of a submodule relative to a module homomorphism. This problem is solved lifting suitably to free modules over polynomial rings. To complete the computation of the conductor one has to represent an integral extension as a finitely presented module; we give an algorithm for this, that at the same time verifies that the extension is integral. — Other algorithms are given that test subring inclusion and birational equivalence. These algorithms are not strictly necessary to perform the conductor computation, but are in some sense connected to the former problem, since if the answer to these tests is negative then the conductor is trivial.

##### MSC:
 13P99 Computational aspects and applications of commutative rings 13B02 Extension theory of commutative rings 13B22 Integral closure of commutative rings and ideals
Full Text:
##### References:
 [1] Bayer, D.A., The division algorithm and the Hilbert scheme, () [2] Gianni, P.; Traverso, C., Subalgebra Gröbner bases, Communicative algebra, Torino, Proceedings of a colloquim in honor of P. salmon, (1990), to appear [3] Kapur, D.; Madlener, K., A completion procedure for computing a canonical basis of a k-subalgebra, () · Zbl 0692.13001 [4] Möller, H.M.; Mora, T., New constructive methods in classical ideal theory, J. algebra, 100, 138-178, (1986) · Zbl 0621.13007 [5] Ollivier, F., Canonical bases: relations with standard bases, finiteness conditions and applications to tame automorphisms, () · Zbl 0734.13015 [6] Shannon, D.; Sweedler, M., Using Gröbner bases to determine algebra membership, split surjective algebra homomorphism and determine birational equivalence, J. symbolic comput., 6, 267-273, (1988) · Zbl 0681.68052 [7] Spear, D., A constructive approach to commutative ring theory, Proceedings 1977 macsyma User’s conference, 369-376, (1977) [8] Traverso, C.; Donati, L., Experimenting the Gröbner basis algorithm with the AIPI system, () [9] Traverso, C., Intersection and inverse algorithms, (), to appear
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.