Infinitely Noncatenary Unique Factorization Domains by Alexandra Bonat '23, Senior Thesis Defense

Infinitely Noncatenary Unique Factorization Domains by Alexandra Bonat ’23, Mathematics Senior Thesis Defense, Wednesday, April 26, 1 – 1:45 pm, North Science Building 113, Wachenheim.

Abstract:  Understanding the structure of the set of prime ideals (the prime spectrum) of a ring is useful to understanding the ring itself. It is reasonable to expect that rings with “nice” properties, for example, unique factorization domains, would have similarly “nice” prime spectra.  However, this is not the case.  It has previously been shown that any finite poset can be embedded in the prime spectrum of a local UFD, i.e. that the prime spectra can be “ugly” at finitely many places.  In this thesis, we construct examples of local UFDs with “infinitely noncatenary” posets embedded in their prime spectra.  We also construct a local UFD that is noncatenary at every height 1 prime ideal, showing that a local UFD can have a prime spectrum that is “ugly” everywhere.