We 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.
Funding statement: The first author received support from a Summer Research Award from St. Louis University’s Office of Research Services. The second author was supported by Polish KBN Grant 1 P03A 032 27.
We thank Arturo Magidin for suggesting Propositions 8.3 and 8.4 and graciously permitting us to include them here. We thank the referee for his or her very helpful and detailed comments, which improved the clarity of the paper. Part of this paper was written while the first author was visiting the Faculty of Computer Science of Bialystok University of Technology. He is deeply grateful for the warm hospitality he received.
 Amitsur S. A., Nil radicals. Historical notes and some new results, Rings, Modules and Radicals (Keszthely 1971), Colloq. Math. Soc. János Bolyai 6, North-Holland, Amsterdam (1973), 47–65. Search in Google Scholar
 Asano K., Über Hauptidealringe mit Kettensatz, Osaka Math. J. 1 (1949), 52–61. Search in Google Scholar
 Bessenrodt C., Brungs H. H. and Törner G., Right chain rings. Part 1, Schriftenreihe des Fachbereiches Mathematik der Universität Duisburg 181 (1990). Search in Google Scholar
 Birkenmeier G. F., Heatherly H. E. and Lee E. K., Completely prime ideals and associated radicals, Ring Theory (Granville 1992), World Scientific Publishing, River Edge (1993), 102–129. Search in Google Scholar
 Birkenmeier G. F., Kim J. Y. and Park J. K., Regularity conditions and the simplicity of prime factor rings, J. Pure Appl. Algebra 115 (1997), no. 3, 213–230. 10.1016/S0022-4049(96)00011-4Search in Google Scholar
 Brungs H. H. and Dubrovin N. I., A classification and examples or rank one chain domains, Trans. Amer. Math. Soc. 355 (2003), no. 7, 2733–2753. 10.1090/S0002-9947-03-03272-0Search in Google Scholar
 Hirano Y., van Huynh D. and Park J. K., On rings whose prime radical contains all nilpotent elements of index two, Arch. Math. (Basel) 66 (1996), no. 5, 360–365. 10.1007/BF01781553Search in Google Scholar
 Jonah D., Rings with the minimum condition for principal right ideals have the maximum condition for principal left ideals, Math. Z. 113 (1970), 106–112. 10.1007/BF01141096Search in Google Scholar
 Kropholler P. H., Linnell P. A. and Moody J. A., Applications of a new K-theoretic theorem to soluble group rings, Proc. Amer. Math. Soc. 104 (1988), no. 3, 675–684. 10.2307/2046771Search in Google Scholar
 Lam T. Y., Corner ring theory: A generalization of Peirce decompositions. I, Algebras, Rings and Their Representations, World Scientific Publishing, Hackensack (2006), 153–182. 10.1142/9789812774552_0011Search in Google Scholar
 Marks G., Mazurek R. and Ziembowski M., A unified approach to various generalizations of Armendariz rings, Bull. Aust. Math. Soc. 81 (2010), no. 3, 361–397. 10.1017/S0004972709001178Search in Google Scholar
 Nicholson W. K. and Zhou Y., Rings in which elements are uniquely the sum of an idempotent and a unit, Glasg. Math. J. 46 (2004), no. 2, 227–236. 10.1017/S0017089504001727Search in Google Scholar
 Puczyłowski E. R., Questions related to Koethe’s nil ideal problem, Algebra and Its Applications, Contemp. Math. 419, American Mathematical Society, Providence (2006), 269–283. 10.1090/conm/419/08010Search in Google Scholar
 Tachikawa H., Lectures on and Rings, Notes by Claus Michael Ringel, Carleton Math. Lecture Notes 1, Department of Mathematics, Carleton University, Ottawa, 1972. 10.1007/BFb0059998Search in Google Scholar
 Tuganbaev A. A., Orders in chain rings, Mat. Zametki 74 (2003), no. 6, 924–933. Search in Google Scholar
© 2016 by De Gruyter