Ore extensions of skew $pi$-Armendariz rings

Abstract

For a ring endomorphism <br />$alpha$ and an $alpha$-derivation $delta$, we introduce a <br />concept, so called skew $pi$-Armendariz ring, that is a <br />generalization of both $pi$-Armendariz rings, <br />and $(alpha,delta)$-compatible skew Armendariz rings. We first <br />observe the basic properties of skew $pi$-Armendariz rings, and <br />extend the class of skew $pi$-Armendariz rings through various ring <br />extensions. We next show that all $(alpha,delta)$-compatible <br />$NI$ rings are skew $pi$-Armendariz, and if a ring $R$ is an <br />$(alpha,delta)$-compatible $2$-$primal$ ring, then the polynomial <br />ring $R[x]$ is skew $pi$-Armendariz.