THE ASCENDING CHAIN CONDITION FOR PRINCIPAL LEFT IDEALS OF SKEW POLYNOMIAL RINGS

Abstract

In this note we study the ascending chain conditions on principal left (resp. right) ideals of the skew polynomial ring $R[x;\alpha,\delta]$. We give a characterization of skew polynomial rings $R[x;\alpha,\delta]$ that are domains and satisfy the ascending chain condition on principal left (resp. right) ideals. We also prove that if $R$ is an $\alpha$-rigid ring that satisfies the ascending chain condition on right annihilators and ascending chain condition on principal right (resp. left) ideals, then the skew polynomial ring $R[x;\alpha,\delta]$ and skew power series ring $R[[x;\alpha]]$ also satisfy the ascending chain condition on principal right (resp. left) ideals.