Acta Mathematica Vietnamica

, Volume 44, Issue 4, pp 977–991 | Cite as

Differential Extensions of Weakly Principally Quasi-Baer Rings

  • Kamal PaykanEmail author
  • Ahmad Moussavi


A ring R is called weakly principally quasi-Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is right s-unital by right semicentral idempotents, which implies that R modulo, the right annihilator of any principal right ideal, is flat. We study the relationship between the weakly p.q.-Baer property of a ring R and those of the differential polynomial extension R[x;δ], the pseudo-differential operator ring R((x− 1;δ)), and also the differential inverse power series extension R[[x− 1;δ]] for any derivation δ of R. Examples to illustrate and delimit the theory are provided.


Differential polynomial ring Pseudo-differential operator ring Differential inverse power series ring (Weakly) p.q.-Baer APP ring AIP ring s-unital ideal 

Mathematics Subject Classification (2010)

16D40 16N60 16S90 16S36 



The authors would like to express their deep gratitude to the referee for a very careful reading of the article, and many valuable comments, which have greatly improved the presentation of the article.

Funding Information

This research was supported by the Iran National Science Foundation: INSF (No: 95004390).


  1. 1.
    Armendariz, E.P.: A note on extensions of Baer and p.p.-rings. J. Austral. Math. Soc. 18, 470–473 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berberian, S.K.: Baer ∗-Rings. Springer, New York (1972)CrossRefzbMATHGoogle Scholar
  3. 3.
    Birkenmeier, G.F., Kim, J.Y., Park, J.K.: On quasi-Baer rings. Algebra and Its Applications, 67–92. Contemp. Math., vol. 259. Am. Math. Soc., Providence (2000)Google Scholar
  4. 4.
    Birkenmeier, G.F., Kim, J.Y., Park, J.K.: Principally quasi-Baer rings. Comm. Algebra 29(2), 639–660 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Birkenmeier, G.F., Kim, J.Y., Park, J.K.: Polynomial extensions of Baer and quasi-Baer rings. J. Pure Appl. Algebra 159(1), 25–42 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Birkenmeier, G.F., Park, J.K.: Triangular matrix representations of ring extensions. J. Algebra 265(2), 457–477 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chase, S.U.: A generalization the ring of triangular matrices. Nagoya Math. J. 18, 13–25 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cheng, Y., Huang, F.K.: A note on extensions of principally quasi-Baer rings. Taiwanese J. Math. 12(7), 1721–1731 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Clark, W.E.: Twisted matrix units semigroup algebras. Duke Math. J. 34, 417–423 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dzhumadil’daev, A.S.: Derivations and central extensions of the Lie algebra of formal pseudo differential operators. Algebra i Anal. 6(1), 140–158 (1994)zbMATHGoogle Scholar
  11. 11.
    Goodearl, K.R.: Centralizers in differential, pseudo differential, and fractional differential operator rings. Rocky Mountain J. Math. 13(4), 573–618 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Goodearl, K.R., Warfield, R.B.: An Introduction to Noncommutative Noetherian Rings. Cambridge University Press, Cambridge (1989)zbMATHGoogle Scholar
  13. 13.
    Hirano, Y.: On annihilator ideals of a polynomial ring over a noncommutative ring. J. Pure Appl. Algebra 168(1), 45–52 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kaplansky, I.: Projections in Banach Algebras. Ann. of Math. (2) 53, 235–249 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kaplansky, I.: Rings of Operators. Benjamin, New York (1968)zbMATHGoogle Scholar
  16. 16.
    Lam, T.Y.: Lectures on modules and rings Graduate Texts in Math, vol. 189. Springer, New York (1999)CrossRefGoogle Scholar
  17. 17.
    Letzter, E.S., Wang, L.: Noetherian skew inverse power series rings. Algebr. Represent. Theory 13(3), 303–314 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Liu, Z.: A note on principally quasi-Baer rings. Comm. Algebra 30(8), 3885–3890 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Liu, Z., Zhao, R.: A generalization of PP-rings and p.q.-Baer rings. Glasg. Math. J. 48(2), 217–229 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Majidinya, A., Moussavi, A., Paykan, K.: Generalized APP-rings. Comm. Algebra 41(12), 4722–4750 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Majidinya, A., Moussavi, A., Paykan, K.: Rings in which the annihilator of an ideal is pure. Algebra Colloq. 22(1), 947–968 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Majidinya, A., Moussavi, A.: Weakly principally quasi-Baer rings. J. Algebra Appl. 15(1), 20 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Manaviyat, R., Moussavi, A., Habibi, M.: Principally quasi-Baer skew power series modules. Comm. Algebra 41(4), 1278–1291 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Manaviyat, R., Moussavi, A.: On annihilator ideals of pseudo-differential operator rings. Algebra Colloq. 22(4), 607–620 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Nasr-Isfahani, A.R., Moussavi, A.: On weakly rigid rings. Glasg. Math. J. 51 (3), 425–440 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Paykan, K., Moussavi, A.: Special properties of diffreential inverse power series rings. J. Algebra Appl. 15(10), 23 (2016)CrossRefzbMATHGoogle Scholar
  27. 27.
    Paykan, K., Moussavi, A.: Study of skew inverse Laurent series rings. J. Algebra Appl. 16(12), 33 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Paykan, K.: Skew inverse power series rings over a ring with projective socle. Czechoslovak Math. J. 67(2), 389–395 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Paykan, K., Moussavi, A.: Primitivity of skew inverse Laurent series rings and related rings. J. Algebra Appl. (2019)
  30. 30.
    Pollingher, A., Zaks, A.: On Baer and quasi-Baer rings. Duke Math. J. 37, 127–138 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Rickart, C.E.: Banach algebras with an adjoint operation. Ann. of Math. (2) 47, 528–550 (1946)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Schur, I.: Uber vertauschbare lineare Differentialausdrucke, Sitzungsber. Berliner Math. Ges. 4, 2–8 (1905)zbMATHGoogle Scholar
  33. 33.
    Small, L.W.: Semihereditary rings. Bull. Am. Math. Soc. 73, 656–658 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Stenström, B.: Rings of Quotients. Springer, New York-Heidelberg (1975)CrossRefzbMATHGoogle Scholar
  35. 35.
    Tominaga, H.: On s-unital rings. Math. J. Okayama Univ. 18(2), 117–134 (1975/76)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Tuganbaev, D.A.: Laurent series rings and pseudo-differential operator rings. J. Math. Sci. (N.Y.) 128(3), 2843–2893 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Pure Mathematics, Faculty of Mathematical SciencesTarbiat Modares UniversityTehranIran

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