ON WEAK ZIP SKEW POLYNOMIAL RINGS

Abstract

We introduce the notion of nil(α, δ)-compatible rings which is a generalization of reduced rings and (α, δ)-compatible rings. In [Ore extensions of weak zip rings, Glasgow Math. J.51 (2009) 525–537] Ouyang introduces the notion of right (respectively, left) weak zip rings and proved that, a ring R is right (respectively, left) weak zip if and only if the skew polynomial ring R[x; α, δ] is right (respectively, left) weak zip, when R is (α, δ)-compatible and reversible. We extend this result to the more general situation that, when R has (α, δ)-condition and quasi-IFP, then nil (R)[x; α, δ] = nil (R[x; α, δ]); and R is right (respectively, left) weak zip if and only if the skew polynomial ring R[x; α, δ] is right (respectively, left) weak zip.