Basic Properties of Unique Factorization Rings and Examples

Introduction

A unique factorization ring (UFR) is a commutative ring with unity in which every non-zero element can be uniquely factored into irreducible elements. This concept generalizes the familiar properties of the ring of integers, $\mathbb{Z}$, where every integer greater than 1 factors into primes, and the uniqueness of this factorization is a cornerstone of number theory. UFRs are crucial in algebraic number theory and