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Forum Mathematicum

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Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 28, Issue 5


Annelidan rings

Greg Marks
  • Corresponding author
  • Department of Mathematics and Computer Science, St. Louis University, St. Louis, MO 63103, United States of America
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/ Ryszard Mazurek
Published Online: 2015-11-17 | DOI: https://doi.org/10.1515/forum-2015-0107


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.

Keywords: Annelidan ring; lineal ring

MSC 2010: 16D25; 16L30; 16N40; 16P60; 16S34; 16D20; 16P40; 16S70; 16U80


  • [1]

    Amitsur A. S., Algebras over infinite fields, Proc. Amer. Math. Soc. 7 (1956), 35–48. Google Scholar

  • [2]

    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. Google Scholar

  • [3]

    Anderson D. D. and Camillo V., Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265–2272. Google Scholar

  • [4]

    Armendariz E. P., A note on extensions of Baer and P.P.-rings, J. Aust. Math. Soc. 18 (1974), 470–473. Google Scholar

  • [5]

    Asano K., Über Hauptidealringe mit Kettensatz, Osaka Math. J. 1 (1949), 52–61. Google Scholar

  • [6]

    Auslander M., Green E. L. and Reiten I., Modules with waists, Illinois J. Math. 19 (1975), 467–478. Google Scholar

  • [7]

    Bass H., Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488. Google Scholar

  • [8]

    Beaumont R. A. and Pierce R. S., Torsion-free rings, Illinois J. Math. 5 (1961), 61–98. Google Scholar

  • [9]

    Bessenrodt C., Brungs H. H. and Törner G., Right chain rings. Part 1, Schriftenreihe des Fachbereiches Mathematik der Universität Duisburg 181 (1990). Google Scholar

  • [10]

    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. Google Scholar

  • [11]

    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. Google Scholar

  • [12]

    Birkenmeier G. F., Kim J. Y. and Park J. K., A characterization of minimal prime ideals, Glasg. Math. J. 40 (1998), no. 2, 223–236. Google Scholar

  • [13]

    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. Google Scholar

  • [14]

    Camillo V. and Nielsen P. P., McCoy rings and zero-divisors, J. Pure Appl. Algebra 212 (2008), no. 3, 599–615. Google Scholar

  • [15]

    Cohn P. M., Reversible rings, Bull. Lond. Math. Soc. 31 (1999), no. 6, 641–648. Google Scholar

  • [16]

    Dubrovin N. I., Rational closures of group rings of left-ordered groups, Mat. Sb. 184 (1993), no. 7, 3–48. Google Scholar

  • [17]

    Everett, Jr. C. J., An extension theory for rings, Amer. J. Math. 64 (1942), 363–370. Google Scholar

  • [18]

    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. Google Scholar

  • [19]

    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. Google Scholar

  • [20]

    Kim N. K., Lee K. H. and Lee Y., Power series rings satisfying a zero divisor property, Comm. Algebra 34 (2006), no. 6, 2205–2218. Google Scholar

  • [21]

    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. Google Scholar

  • [22]

    Lam T. Y., Lectures on Modules and Rings, Grad. Texts in Math. 189, Springer, New York, 1999. Google Scholar

  • [23]

    Lam T. Y., A First Course in Noncommutative Rings, 2nd ed., Grad. Texts in Math. 131, Springer, New York, 2001. Google Scholar

  • [24]

    Lam T. Y., Corner ring theory: A generalization of Peirce decompositions. I, Algebras, Rings and Their Representations, World Scientific Publishing, Hackensack (2006), 153–182. Google Scholar

  • [25]

    Lee Y., Huh C. and Kim H. K., Questions on 2-primal rings, Comm. Algebra 26 (1998), no. 2, 595–600. Google Scholar

  • [26]

    Marks G., A taxonomy of 2-primal rings, J. Algebra 266 (2003), no. 2, 494–520. Google Scholar

  • [27]

    Marks G. and Mazurek R., On annelidan, distributive, and Bézout rings, in preparation. Google Scholar

  • [28]

    Marks G. and Mazurek R., Rings with linearly ordered right annihilators, submitted. Google Scholar

  • [29]

    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. Google Scholar

  • [30]

    Mesyan Z., The ideals of an ideal extension, J. Algebra Appl. 9 (2010), no. 3, 407–431. Google Scholar

  • [31]

    Mewborn A. C. and Winton C. N., Orders in self-injective semi-perfect rings, J. Algebra 13 (1969), 5–9. Google Scholar

  • [32]

    Morita K., On S-rings in the sense of F. Kasch, Nagoya Math. J. 27 (1966), 687–695. Google Scholar

  • [33]

    Nicholson W. K. and Yousif M. F., Principally injective rings, J. Algebra 174 (1995), no. 1, 77–93. Google Scholar

  • [34]

    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. Google Scholar

  • [35]

    Osofsky B. L., Loewy length of perfect rings, Proc. Amer. Math. Soc. 28 (1971), 352–354. Google Scholar

  • [36]

    Posner E. C., Left valuation rings and simple radical rings, Trans. Amer. Math. Soc. 107 (1963), 458–465. Google Scholar

  • [37]

    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. Google Scholar

  • [38]

    Schröder M., Über N. I. Dubrovins Ansatz zur Konstruktion von nicht vollprimen Primidealen in Kettenringen, Results Math. 17 (1990), no. 3–4, 296–306. Google Scholar

  • [39]

    Shin G., Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43–60. Google Scholar

  • [40]

    Snider R. L., On the singular ideal of a group algebra, Comm. Algebra 4 (1976), no. 11, 1087–1089. Google Scholar

  • [41]

    Tachikawa H., Lectures on QF-3 and QF-1 Rings, Notes by Claus Michael Ringel, Carleton Math. Lecture Notes 1, Department of Mathematics, Carleton University, Ottawa, 1972. Google Scholar

  • [42]

    Tuganbaev A. A., Distributive rings, uniserial rings of fractions, and endo-Bezout modules, J. Math. Sci. (N. Y.) 114 (2003), no. 2, 1185–1203. Google Scholar

  • [43]

    Tuganbaev A. A., Orders in chain rings, Mat. Zametki 74 (2003), no. 6, 924–933. Google Scholar

About the article

Received: 2015-06-04

Revised: 2015-10-08

Published Online: 2015-11-17

Published in Print: 2016-09-01

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.

Citation Information: Forum Mathematicum, Volume 28, Issue 5, Pages 923–941, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0107.

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