Annelidan rings

Abstract

AbstractWe introduce the class of right annelidan rings, defined by the property that any annihilator right ideal of the ring is comparable with every right ideal of the ring. This class is a common generalization of the classes of domains and right uniserial rings. We obtain results on the structure of right annelidan rings; in particular, we show that all right annelidan rings are Armendariz. We study the relationships between right annelidan rings, chain conditions, and 2-primal rings. For the class of right annelidan rings, we prove a version of the Hopkins–Levitzki Theorem for principal right ideals. We characterize right annelidan group algebras, obtaining a classification that is complete if the zero-divisor problem has a positive solution.