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University of Louisville

1. Gipson, Ryan H. Factorization in integral domains.

Degree: PhD, 2018, University of Louisville

We investigate the atomicity and the AP property of the semigroup rings F[X; M], where F is a field, X is a variable and M is a submonoid of the additive monoid of nonnegative rational numbers. In this endeavor, we introduce the following notions: essential generators of M and elements of height (0, 0, 0, . . .) within a cancellative torsion-free monoid Γ. By considering the latter, we are able to determine the irreducibility of certain binomials of the form Xπ − 1, where π is of height (0, 0, 0, . . .), in the monoid domain. Finally, we will consider relations between the following notions: M has the gcd/lcm property, F[X; M] is AP, and M has no elements of height (0, 0, 0, . . .). Advisors/Committee Members: Kulosman, Hamid, Hill, Aaron, Hill, Aaron, Li, Jinjia, Seif, Steve, Brown, David N..

Subjects/Keywords: commutative algebra; integral domains; monoid domains; factorization; Algebra

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APA (6th Edition):

Gipson, R. H. (2018). Factorization in integral domains. (Doctoral Dissertation). University of Louisville. Retrieved from 10.18297/etd/3056 ; https://ir.library.louisville.edu/etd/3056

Chicago Manual of Style (16th Edition):

Gipson, Ryan H. “Factorization in integral domains.” 2018. Doctoral Dissertation, University of Louisville. Accessed July 03, 2020. 10.18297/etd/3056 ; https://ir.library.louisville.edu/etd/3056.

MLA Handbook (7th Edition):

Gipson, Ryan H. “Factorization in integral domains.” 2018. Web. 03 Jul 2020.

Vancouver:

Gipson RH. Factorization in integral domains. [Internet] [Doctoral dissertation]. University of Louisville; 2018. [cited 2020 Jul 03]. Available from: 10.18297/etd/3056 ; https://ir.library.louisville.edu/etd/3056.

Council of Science Editors:

Gipson RH. Factorization in integral domains. [Doctoral Dissertation]. University of Louisville; 2018. Available from: 10.18297/etd/3056 ; https://ir.library.louisville.edu/etd/3056


University of Tennessee – Knoxville

2. Lynch, Benjamin Ryan. Elasticity of Krull Domains with Infinite Divisor Class Group.

Degree: 2010, University of Tennessee – Knoxville

The elasticity of a Krull domain R is equivalent to the elasticity of the block monoid B(G,S), where G is the divisor class group of R and S is the set of elements of G containing a height-one prime ideal of R. Therefore the elasticity of R can by studied using the divisor class group. In this dissertation, we will study infinite divisor class groups to determine the elasticity of the associated Krull domain. The results will focus on the divisor class groups Z, Z(p infinity), Q, and general infinite groups. For the groups Z and Z(p infinity), it has been determined which distributions of the height-one prime ideals will make R a half-factorial domain (HFD). For the group Q, certain distributions of height-one prime ideals are proven to make R an HFD. Finally, the last chapter studies general infinite groups and groups involving direct sums with Z. If certain conditions are met, then the elasticity of these divisor class groups is the same as the elasticity of simpler divisor class groups.

Subjects/Keywords: Krull domains; divisor class group; elasticity; half-factorial domain; block monoid; non-unique factorization; Algebra

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Lynch, B. R. (2010). Elasticity of Krull Domains with Infinite Divisor Class Group. (Doctoral Dissertation). University of Tennessee – Knoxville. Retrieved from https://trace.tennessee.edu/utk_graddiss/821

Chicago Manual of Style (16th Edition):

Lynch, Benjamin Ryan. “Elasticity of Krull Domains with Infinite Divisor Class Group.” 2010. Doctoral Dissertation, University of Tennessee – Knoxville. Accessed July 03, 2020. https://trace.tennessee.edu/utk_graddiss/821.

MLA Handbook (7th Edition):

Lynch, Benjamin Ryan. “Elasticity of Krull Domains with Infinite Divisor Class Group.” 2010. Web. 03 Jul 2020.

Vancouver:

Lynch BR. Elasticity of Krull Domains with Infinite Divisor Class Group. [Internet] [Doctoral dissertation]. University of Tennessee – Knoxville; 2010. [cited 2020 Jul 03]. Available from: https://trace.tennessee.edu/utk_graddiss/821.

Council of Science Editors:

Lynch BR. Elasticity of Krull Domains with Infinite Divisor Class Group. [Doctoral Dissertation]. University of Tennessee – Knoxville; 2010. Available from: https://trace.tennessee.edu/utk_graddiss/821

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