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.