Commutative Polynomial Rings which are Principal Ideal Rings

Abstract

A well-known result by Zariski and Samuel asserts that a commutative principal ideal ring is a direct sum of finitely many principal ideal domains and Artinian chain rings. Based on this result, it is shown, among other things, that a commutative polynomial ring R[x] is a principal ideal ring if and only if R is a finite direct sum of fields.