Research Overview
My research into factorization theory continues in a rich tradition of investigation. Broadly speaking, I study how elements of a ring break down (or fail to break down) into irreducible elements. Often, this work takes place in the context of order in a number field, which are specific types of rings of interest to the field of algebraic number theory. I also investigate how one can leverage certain relationships between a ring and its subrings to provide additional insight into their structures.
Throughout my published works, I have sought to answer, among others, the following broad questions:
Given an order R in a number field K, how can we determine the elasticity of R?
What factorization information can we determine about the ring of polynomials R[x] or the ring of formal power series R[[x]] based on R?
How can we determine when a subring R of a larger ring T is associated, ideal-preserving, or locally associated (defined in [4]), and what do these properties tell us about R, T, and other related rings?
My work thus far has produced interesting results, including:
When R is an order whose conductor ideal is prime in R and principal in its integral closure, we can determine the elasticity of R using the S-relative Davenport constant. [7]
If R is an order in a number field with radical conductor ideal I, then R is an associated order if and only if the elasticity of R is the same as that of its integral closure. [8] and [6]
An order in a number field is associated if and only if it is both ideal-preserving and locally associated. [5]
The ring of formal power series R[[x]] over a ring of algebraic integers R is an HFD if and only if R is an HFD. [10]