The study of rings plays a vital role within abstract algebra, as well as in other mathematics topics. Many disciplines draw from abstract algebra and ring theory: for example, number theory focuses on the ring of integers, and algebraic geometry draws from techniques used in ring theory.
In 1993, Raymond Heitmann published an article titled “Characterization of Completions of Unique Factorization Domains”. Despite being published a relatively long time ago, Heitmann’s article is by no means easily accessible. Thus, this paper seeks to explore just a small part of Heitmann’s article. The first chapter seeks to give the reader a solid foundation for rings, ideals, quotient rings, and homomorphisms. Following that, the second chapter will use those foundations to define localizations, discrete valuations, and generalized completions. Finally, the last chapter will focus on making sense of part of Heitmann’s proof of Theorem 1 in his article. The paper concludes with a consideration of the further work needed to further understand the article, as well as references to similar articles that cite “Characterization of Completions of Unique Factorization Domains”.
Richardson, Lily, "Rings & Things: Localization, Discrete Valuations, and Completion in Ring Theory" (2020). Senior Independent Study Theses. Paper 8960.
algebra, Abstract algebra, completion, localization, discrete valuation, principal ideal domain, laurent series, power series, unique factorization
Bachelor of Arts
Senior Independent Study Thesis
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