Abstract.Results on the quasi-cyclicity of the Gray map image of a class of codes defined over the Galois ring GR(p 2 , m) are given. These results generalize some appearing in [8] for codes over the ring Z Z/p 2 Z Z of integers modulo p 2 (p a prime). The ring of (truncated) Witt vectors is a useful tool in proving the main results.
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.
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