Strong n-generators in some one-dimensional domains
1998 (English)Doctoral thesis, comprehensive summary (Other academic)
Let R be a commutative Noetherian one-dimensional domain such that each ideal in R can be generated by n elements. A strong n-generator in R is an element r which can be chosen as one of n generators for each ideal in which it is contained. If moreover r is contained in some ideal requiring n generators, then r is said to be a proper strong n-generator. We prove that if R is two-generated, then r is a strong two-generator if and only if r doesn't belong to Mˆ2 for any non-invertible maximal ideal M of R. Let S be an index set such that for each i in S, M(i) is a maximal ideal such that the localization of R at M(i) possesses an ideal requiring n generators. We show that there is a unique biggest M(i)-primary ideal I(i) in R requiring n generators. We prove that r is a strong n-generator if and only if r doesn't belong to I(i)M(i) for any i in S. Let R' be the integral closure of R. Suppose moreover that the conductor C of R' in R is non-zero. We prove that if n > 2, then r is a proper strong n-generator if and only if r belongs to the complement of I(i)M(i) in I(i) for some i in S. And if n = 2, then r is a proper strong two-generator if and only if r belongs to the complement of Mˆ2 in M for some non-invertible maximal ideal M of R or r belongs to some non-principal invertible maximal ideal of R. We prove that if C requires n generators, then R is n-generated and the extension of C to the localization of R at M(i) for some i in S requires n generators.
Suppose moreover that R is a Cohen-Macaulay ring, R possesses a superficial element and R' is a local domain. Then (R,M) is a local domain. Let B(M) be the blowing-up ring of R at M. We give an algorithm for determining the conductor J of B(M) in R. We determine the reduction exponent of M for a semigroup ring and give a sufficient condition for J not being a power of M.
Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 1998. , 20 p.
Research subject Mathematics
IdentifiersURN: urn:nbn:se:su:diva-45754ISBN: 91-7153-751-1OAI: oai:DiVA.org:su-45754DiVA: diva2:369523
1999-05-20, Sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 09:00 (English)
Anderson, Daniel D, Professor
Härtill fyra uppsatser.2010-11-102010-11-102010-11-11Bibliographically approved