Ore extensions of skew pipi-Armendariz rings

For a ring endomorphism alphaalpha and an alphaalpha-derivation deltadelta, we introduce a concept, so called skew pipi-Armendariz ring, that is a generalization of both pipi-Armendariz rings, and (alpha,delta)(alpha,delta)-compatible skew Armendariz rings. We first observe the basic properties of skew pipi-Armendariz rings, and extend the class of skew pipi-Armendariz rings through various ring extensions. We next show that all (alpha,delta)(alpha,delta)-compatible NINI rings are skew pipi-Armendariz, and if a ring RR is an (alpha,delta)(alpha,delta)-compatible 22-primalprimal ring, then the polynomial ring R[x]R[x] is skew pipi-Armendariz.