On alpha-nilpotent elements and alpha-Armendariz rings

摘要

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 alpha-Armendariz ring, where a is a ring monomorphism. In this paper, we investigate the structure of the set of alpha-nilpotent elements in a-Armendariz rings and introduce an alpha-nil-Armendariz ring. We examine the set of (alpha) over bar -nilpotent elements in a skew polynomial ring R[x;alpha], where (alpha) over bar is the monomorphism induced by the monomorphism a of an a-Armendariz ring R. We prove that every polynomial with a-nilpotent coefficients in a ring R is (alpha) over bar -nilpotent when R is of bounded index of alpha-nilpotency, and moreover, R is shown to be alpha-nil-Armendariz in this situation. We also characterize the structure of the set of alpha-nilpotent elements in alpha-nil-Armendariz rings, and investigate the relations between alpha-(nil-) Armendariz property and other standard ring theoretic properties.