Year: 2024
Type: article
Abstract: Let D be a Euclidean domain, with fraction field K. Let R(D) be the subring of K generated by the reciprocals of the nonzero elements of D. The main theorem states that if $$R(D) \ne K$$ R ( D ) ≠ K , then R(D) is a rank 1 discrete valuation ring that contains a field consisting of the units of D along with 0. Along the way, it is shown that R(D) contains all fractions of the form a/b, where $$a,b \in D \setminus \{0\}$$ a , b ∈ D \ { 0 } and a has Euclidean value no larger than that of b. Connections are also made to ideas from medieval Italian mathematics.
Source: Ricerche di Matematica
Author: Neil Epstein
Institution: George Mason University
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4405776144