Multiplicative lattices and the integral closure operation

dc.contributor.advisorJohnson, Johnny A.
dc.contributor.committeeMemberBatten, George W., Jr.
dc.contributor.committeeMemberBrown, Dennison R.
dc.contributor.committeeMemberHausen, Jutta
dc.contributor.committeeMemberVobach, Arnold R.
dc.creatorBecerra, Linda
dc.description.abstractIn 1969, J. P. Lediaev introduced the concept of the integral closure of an element in a multiplicative lattice. He showed that in the lattice of ideals of a Noetherian ring, the integral closure of an ideal coincides with its integral closure as an element of the lattice. In this dissertation, we study the integral closure operation in multiplicative lattices. In particular, attention is focused on integrally closed principal elements in multiplicative lattices, and on multiplicative lattices in which each nonzero principal element is integrally closed; these will be called tidy lattices. It is shown that integrally closed principal nonzero divisors can be characterized in terms of their prime divisors in multiplicative lattices which are Noether lattices. Then, to study tidy lattices, the concept of a direct sum of a multiplicative lattice is introduced. It is shown that tidy Noether lattices which are not domains and satisfy the union condition on primes can be characterized in terms of their minimal direct summands. Finally, the integral closure operation is used to study the prime divisors of all large powers of a fixed element of a Noether lattice.
dc.description.departmentMathematics, Department of
dc.format.digitalOriginreformatted digital
dc.rightsThis item is protected by copyright but is made available here under a claim of fair use (17 U.S.C. Section 107) for non-profit research and educational purposes. Users of this work assume the responsibility for determining copyright status prior to reusing, publishing, or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires express permission of the copyright holder.
dc.subjectClosure operators
dc.subjectLattice theory
dc.titleMultiplicative lattices and the integral closure operation
dc.type.genreThesis of Natural Sciences and Mathematics, Department of of Houston of Philosophy


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