Document Type : Final File


Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-134, Tehran, Islamic Republic of Iran


 Letbe a ring with an endomorphism and an -derivationAntoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. In this paper we introduce and investigate the notion of nil--compatible rings. The class of nil--compatible rings are extended through various ring extensions and many classes of nil--compatible rings are constructed. We also prove that, if  is nil--compatible and nil-Armendariz ring of power series type with  nilpotent, then . We show that, if  is a nil-Armendariz ring of power series type, with  nilpotent and nil--compatible ring, then  As a consequence, several known results are unified and extended to the more general setting. Also examples are provided to illustrate our results.