Some Divisibility Properties in Ring of Polynomials over a Unique Factorization Domain

TL;DR

This paper explores divisibility properties in polynomial rings over UFDs, providing criteria for divisibility, characterizations of D-rings, and results on the distribution of prime elements in coefficient domains.

Contribution

It introduces new divisibility criteria via polynomial evaluation and offers alternative characterizations of D-rings, advancing understanding of prime elements in coefficient domains.

Findings

Criteria for polynomial divisibility using evaluation methods

Conditions under which coefficient domains have infinitely many primes

Alternative characterizations of D-rings

Abstract

Using polynomial evaluation, we give some useful criteria to answer questions about divisibility of polynomials. This allows us to develop interesting results concerning the prime elements in the domain of coefficients. In particular, it is possible to prove that under certain conditions, the domain of coefficients must have infinitely many prime elements. We give alternative characterizations for D-rings and present various examples.