Algebras and Representation Theory

, Volume 21, Issue 2, pp 359–373 | Cite as

Criteria for a Ring to have aLeft Noetherian Largest Left Quotient Ring

Open Access
Article

Abstract

Criteria are given for a ring to have a left Noetherian largest left quotient ring. It is proved that each such a ring has only finitely many maximal left denominator sets. An explicit description of them is given. In particular, every left Noetherian ring has only finitely many maximal left denominator sets.

Keywords

Goldie’s Theorem The left quotient ring of a ring The largest left quotient ring of a ring A maximal left denominator set The left localization radical of a ring An Ore set A left denominator set The prime radical 

Mathematics Subject Classification (2010)

16P50 16P60 16P20 16U20 

References

  1. 1.
    Bavula, V.V.: The algebra of integro-differential operators on an affine line and its modules. J. Pure Appl. Algebra 217, 495–529 (2013). arXiv:math.RA.1011.2997 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bavula, V.V.: New criteria for a ring to have a semisimple left quotient ring. J. Alg. Appl. 6(6), 28 (2015). doi: 10.1142/S0219498815500905. arXiv:math.RA:1303.0859 MathSciNetMATHGoogle Scholar
  3. 3.
    Bavula, V.V.: The largest strong left quotient ring of a ring. J. Algebra 439, 1–32 (2015). arXiv:math.RA.1310.1077 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bavula, V.V.: The largest left quotient ring of a ring. Commun. Algebra 44(18), 3219–3261 (2016). arXiv:math.RA.1101.5107 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bavula, V.V.: Left localizations of left Artinian rings. J. Alg. Appl. 15 (9), 1650165, 38 (2016). doi: 10.1142/S0219498816501656. arXiv:math.RA:1405.0214 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bavula, V.V.: Criteria for a ring to have a left Noetherian left quotient ring. arXiv:math.RA.1508.03798
  7. 7.
    Birkenmeier, G.F., Park, J.K., Rizvi, S.T.: Ring hulls and applications. J. Algebra 304, 633–665 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chatters, A.W., Hajarnavis, C.R.: Quotient rings of Noetherian module finite rings. Proc. AMS 121(2), 335–341 (1994)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jategaonkar, A.V.: Localization in noetherian rings. Londom Mathematics Society. LMS 98, Cambridge University Press, J. Pure Appl. Alg., to appear (1986)Google Scholar
  10. 10.
    McConnell, J.C., Robson, J.C.: Noncommutative Noetherian rings. With the cooperation of L. W. Small. Revised edition. Graduate Studies in Mathematics, 30, p 636. American Mathematical Society, Providence, RI (2001)MATHGoogle Scholar
  11. 11.
    Michler, G., Müller, B.: T he maximal regular Ore set of a Noetherian ring. Arch. Math. 43, 218–223 (1984)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Stafford, J.T.: Noetherian full quotient rings. Proc. Lond. Math. Soc. (3) 44, 385–404 (1982)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Stenström, B.: Rings of quotients. Springer-Verlag, Berlin, Heidelberg New York (1975)CrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of SheffieldHicks BuildingUK

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