On α-nilpotent elements and α-Armendariz rings

Abstract

Antoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced the concept of nil-Armendariz property as a generalization. Hong et al. studied Armendariz property on skew polynomial rings and introduced the notion of an α-Armendariz ring, where α is a ring monomorphism. In this paper, we investigate the structure of the set of α-nilpotent elements in α-Armendariz rings and introduce an α-nil-Armendariz ring. We examine the set of [Formula: see text]-nilpotent elements in a skew polynomial ring R[x;α], where [Formula: see text] is the monomorphism induced by the monomorphism α of an α-Armendariz ring R. We prove that every polynomial with α-nilpotent coefficients in a ring R is [Formula: see text]-nilpotent when R is of bounded index of α-nilpotency, and moreover, R is shown to be α-nil-Armendariz in this situation. We also characterize the structure of the set of α-nilpotent elements in α-nil-Armendariz rings, and investigate the relations between α-(nil-)Armendariz property and other standard ring theoretic properties.