Israel Journal of Mathematics

, Volume 216, Issue 1, pp 415–440

Rings with linearly ordered right annihilators

Article

Abstract

We introduce the class of lineal rings, defined by the property that the lattice of right annihilators is linearly ordered. We obtain results on the structure of these rings, their ideals, and important radicals; for instance, we show that the lower and upper nilradicals of these rings coincide. We also obtain an affirmative answer to the Köthe Conjecture for this class of rings. We study the relationships between lineal rings, distributive rings, Bézout rings, strongly prime rings, and Armendariz rings. In particular, we show that lineal rings need not be Armendariz, but they fall not far short.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. A. Amitsur, Nil radicals. Historical notes and some new results, in Rings, modules and radicals (Proc. Internat. Colloq., Keszthely, 1971), North-Holland, Amsterdam, 1973, pp. 47–65. Colloq. Math. Soc. János Bolyai, Vol. 6.MATHGoogle Scholar
  2. [2]
    D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), 2265–2272.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    V. A. Andrunakievič, Radicals of associative rings. I, Mat. Sb. N.S. 44(86) (1958), 179–212.MathSciNetGoogle Scholar
  4. [4]
    E. P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc. 18 (1974), 470–473.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    G. M. Bergman, The diamond lemma for ring theory, Adv. in Math. 29 (1978), 178–218.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    C. Bessenrodt, H. H. Brungs and G. Törner, Right chain rings, part 1, Schriftenreihe des Fachbereichs Mathematik der Universität Duisburg, Vol. 181, 1990.MATHGoogle Scholar
  7. [7]
    H. H. Brungs, Rings with a distributive lattice of right ideals, J. Algebra 40 (1976), 392–400.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    H. H. Brungs and N. I. Dubrovin, A classification and examples of rank one chain domains, Trans. Amer. Math. Soc. 355 (2003), 2733–2753.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    V. Camillo, Distributive modules, J. Algebra 36 (1975), 16–25.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    V. Camillo and P. P. Nielsen, McCoy rings and zero-divisors, J. Pure Appl. Algebra 212 (2008), 599–615.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    M. Ferrero and G. Törner, On the ideal structure of right distributive rings, Comm. Algebra 21 (1993), 2697–2713.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    K. R. Goodearl and D. Handelman, Simple self-injective rings, Comm. Algebra 3 (1975), 797–834.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    D. Handelman and J. Lawrence, Strongly prime rings, Trans. Amer. Math. Soc. 211 (1975), 209–223.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168 (2002), 45–52.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Y. Hirano, D. van Huynh and J. K. Park, On rings whose prime radical contains all nilpotent elements of index two, Arch. Math. (Basel) 66 (1996), 360–365.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    C. Huh, Y. Lee and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), 751–761.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    C. U. Jensen, On characterizations of Prüfer rings, Math. Scand. 13 (1963), 90–98.MathSciNetMATHGoogle Scholar
  18. [18]
    N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), 477–488.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, Vol. 189, Springer-Verlag, New York, 1999.CrossRefGoogle Scholar
  20. [20]
    T. Y. Lam, A first course in noncommutative rings, second ed., Graduate Texts in Mathematics, Vol. 131, Springer-Verlag, New York, 2001.CrossRefGoogle Scholar
  21. [21]
    T.-K. Lee and T.-L. Wong, On Armendariz rings, Houston J. Math. 29 (2003), 583–593 (electronic).MathSciNetMATHGoogle Scholar
  22. [22]
    G. Marks and R. Mazurek, Annelidan rings, Forum Math., to appear.Google Scholar
  23. [23]
    G. Marks, R. Mazurek and M. Ziembowski, A unified approach to various generalizations of Armendariz rings, Bull. Aust. Math. Soc. 81 (2010), 361–397.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    R. Mazurek, Distributive rings with Goldie dimension one, Comm. Algebra 19 (1991), 931–944.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    R. Mazurek and E. R. Puczyłowski, On nilpotent elements of distributive rings, Comm. Algebra 18 (1990), 463–471.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    E. R. Puczyłowski, Questions related to Koethe’s nil ideal problem, in Algebra and its applications, Contemp. Math., Vol. 419, Amer. Math. Soc., Providence, RI, 2006, pp. 269–283.CrossRefGoogle Scholar
  27. [27]
    R. A. Rubin, Absolutely torsion-free rings, Pacific J. Math. 46 (1973), 503–514.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    W. Stephenson, Modules whose lattice of submodules is distributive, Proc. London Math. Soc. (3) 28 (1974), 291–310.MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    A. A. Tuganbaev, Distributive rings, uniserial rings of fractions, and endo-Bezout modules, J. Math. Sci. (N. Y.) 114 (2003), 1185–1203, Algebra, 22.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    P. Vámos, Finitely generated Artinian and distributive modules are cyclic, Bull. London Math. Soc. 10 (1978), 287–288.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    H.-P. Yu, On quasi-duo rings, Glasgow Math. J. 37 (1995), 21–31.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceSt. Louis UniversitySt. LouisUSA
  2. 2.Faculty of Computer ScienceBialystok University of TechnologyBiałystokPoland

Personalised recommendations