Date on Master's Thesis/Doctoral Dissertation
Applied and Industrial Mathematics, PhD
Committee Co-Chair (if applicable)
Brown, David N.
commutative algebra; integral domains; monoid domains; factorization
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, . . .).
Gipson, Ryan H., "Factorization in integral domains." (2018). Electronic Theses and Dissertations. Paper 3056.