When is R[x] a principal ideal ring?

Abstract

Abstract. Because of its interesting applications in coding theory, cryptography, and algebraic combinatorics, in recent decades a lot of attention has been paid to the algebraic structure of the ring of polynomials R [x], where R is a finite commutative ring with identity. Motivated by this popularity, in this paper we determine when R[x] is a principal ideal ring. In fact, we prove that R[x] is a principal ideal ring if and only if R is a finite direct product of finite fields. Keywords: Principal ideal ring, polynomial ring, finite rings. MSC2010: 13F10, 13F20, 16P10, 13C05. ¿Cuándo R[x] es un anillo de ideales principales?Resumen. Debido a sus interesantes aplicaciones en teoría de códigos, criptografía y combinatoria algebraica, en décadas recientes se ha incrementado la atención en la estructura algebraica del anillo de polinomios R[x], donde R es un anillo conmutativo finito con identidad. Motivados por esta popularidad, en este artículo determinamos cuándo R[x] es un anillo de ideales principales. De hecho, demostramos que R[x] es un anillo de ideales principales, si y sólo si, R es un producto directo finito de campos finitos. Palabras clave: Anillo de ideales principales, anillo de polinomios, anillos finitos.

Annihilator ideals of indecomposable modules of finite-dimensional pointed Hopf algebras of rank one

“…Afterwards, Decruyenaere and Jespers [8] investigated when a commutative ring with identity graded by an abelian group is a principal ideal ring. Chimal-Dzul and López-Andrade [7] studied the polynomial ring R[x], where R is a finite commutative ring with identity. They proved that R[x] is a principal ideal ring if and only if R is a finite direct product of finite fields.…”