Acta Mathematica Hungarica

, Volume 156, Issue 1, pp 91–101 | Cite as

Clean-like properties in pullbacks and amalgamation rings

  • A. MimouniEmail author


We study the transfer of the notions of clean rings, weakly clean rings, almost clean rings and unit regular rings to various context of pullbacks and amalgamations. Our attempt is to present original and new classes of these rings.

Key words and phrases

clean ring weakly clean ring almost clean ring unit regular ring pullbacks amalgamation 

Mathematics Subject Classification

primary 13A15 13F05 secondary 13G05 13F30 


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The author expresses his sincere thanks to the referee for helpful suggestions and comments, which have greatly improved this paper.


  1. 1.
    M. S. Ahn, Weakly clean and almost clean rings, Ph.D. thesis, University of Iowa (2003).Google Scholar
  2. 2.
    Ahn M.S., Anderson D.D.: Weakly clean rings and almost clean rings. Rocky Mountain J. Math., 36, 783–798 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anderson D.D., Camillo A.: Commutative rings whose elements are a sum of a unit and idempotent. Comm. Algebra 30, 3327–3336 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    D’Anna M.: A construction of Gorenstein rings. J. Algebra 306, 507–519 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. D’Anna, C. A. Finocchiaro and M. Fontana, Amalgamated algebras along an ideal, in : Commutative Algebra and its Applications, Walter de Gruyter (Berlin, 2009), pp. 241–252.Google Scholar
  6. 6.
    D’Anna M., Fontana M.: The amalgamated duplication of a ring along a multiplicative-canonical ideal. Ark. Mat. 45, 241–252 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D’Anna M., Fontana M.: An amalgamated duplication of a ring along an ideal, the basic properties. J. Algebra Appl., 6, 443–459 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. F. Atiyah and I. G. MacDonald, Introduction to Commutatice Algebra, Addison-Wesley (1969).Google Scholar
  9. 9.
    Bastida E., Gilmer R.: Overrings and divisorial ideals of rings of the form D+M. Michigan Math. J., 20, 79–95 (1992)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Boisen M., Sheldon P.B.: CPI-extension: Overrings of integral domains with special prime spectrum. Canad. J. Math., 29, 722–737 (1977)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Camillo V.P., Khurana D.: A characterization of unit regular rings. Comm. Algebra, 29, 2293–2295 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chin A. Y. M., Qua K. T.: A note on weakly clean rings. Acta Math. Hungar., 132, 113–116 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chhiti M., Mahdou N., Tamekkante M.: Clean property in amalgamated algebras along an ideal. Hacettepe J. Math. Stat., 44, 41–49 (2015)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fontana M.: Topologically defined classes of commutative rings. Ann. Mat. Pura Appl., 123, 331–355 (1980)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fontana M., Gabelli S.: On the class group and local class group of a pullback. J. Algebra, 181, 803–835 (1996)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gabelli S., Houston E.: Coherentlike conditions in pullbacks. Michigan Math. J. 44, 99–122 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    McGovern W.W.: Clean semiprime f-rings with bounded inversion. Comm. Algebra, 31, 3295–3304 (2003)MathSciNetCrossRefGoogle Scholar
  18. 18.
    M. Nagata, Local Rings, Interscience (New York, 1962).Google Scholar
  19. 19.
    Nicholson W.K.: Lifting idempotents and exchange rings. Trans. Amer. Math. Soc., 229, 269–278 (1977)MathSciNetCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsKing Fahd University of Petroleum & MineralsDhahranSaudi Arabia

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