Date on Master's Thesis/Doctoral Dissertation

8-2018

Document Type

Doctoral Dissertation

Degree Name

Ph. D.

Department

Mathematics

Degree Program

Applied and Industrial Mathematics, PhD

Committee Chair

Kulosman, Hamid

Committee Co-Chair (if applicable)

Hill, Aaron

Committee Member

Hill, Aaron

Committee Member

Li, Jinjia

Committee Member

Seif, Steve

Committee Member

Brown, David N.

Author's Keywords

commutative algebra; integral domains; monoid domains; factorization

Abstract

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, . . .).

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