Rings with the annihilator condition and their extensions

Abstract

Let R be an associative unital ring and let α be an endomorphism of R. In this article, we study rings having the annihilator condition (a.c.) and some related rings. Also, many polynomial extensions of R having the condition (a.c.) are stud-ied. In particular, α-skew quasi-Armendariz rings and α-(sps) quasi-Armendariz rings. We prove that this class of rings have always the property (a.c.) on the left but not necessary on the right. Furthermore, we investigate the transmission of the property (a.c) from a ring R to R[x, α] and R[[x, α]].