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Licensed Unlicensed Requires Authentication Published by De Gruyter November 17, 2015

Annelidan rings

Greg Marks and Ryszard Mazurek
From the journal Forum Mathematicum

Abstract

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.


Communicated by Manfred Droste


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.

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Received: 2015-6-4
Revised: 2015-10-8
Published Online: 2015-11-17
Published in Print: 2016-9-1

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