Factorization in integral domains.

Abstract

for all their time and suggestions in making this dissertation a reality. I would also extend my appreciation to the faculties of both the University of Louisville and Murray State University; in particular, I would like to extend a very special thanks to Dr. Timothy Schroeder who inspired my mathematical aspirationas as a freshman at Murray State University and to Dr. Thomas Riedel, Department Chair of the University ofLouisville, for his continuing support and assistance beyond his prescribed duties; his genuine interest in students' success has neither gone unnoticed nor unappreciated.Naturally, there are many scholars, professors, colleagues, friends, and family who have all played a unique, and invaluable, role in my achievement and are worthy of due honor. In particular, I would give special honor to my "Mama" who has believed in me before I was born, to my Dad who planted a seed of curiosity and wonder within me, and most especially to my beautiful wife Michele whose loving patience is a pillar of my life.iv Finally, and above all else, I would like to acknowledge the grace and blessings of the Lord Jesus Christ by whose providences I have been blessed to know those mentioned herein, as well as, to have produced this dissertation to what I hope isWe 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 X π − 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