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Some results on skew Hurwitz series rings

  • Abdolreza Tehranchi
  • Kamal PaykanEmail author
Article
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Abstract

A ring is quasi-Baer (respectively, Baer) in case the right annihilator of every ideal (respectively, subset) is generated by an idempotent, as a right ideal. In this article, we investigate on the relationship between the quasi-Baerness, Baerness, semiprimitivity, simplicity and NI properties of a ring R, and its skew Hurwitz series ring \((HR, \alpha )\), where R is a ring equipped with an endomorphism \(\alpha \).

Keywords

Skew Hurwitz series ring (Quasi)  Baer ring Semiprimitive ring Simple ring NI-ring 

Mathematics Subject Classification

16S34 16S35 16S36 16W60 16D40 
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Notes

Acknowledgements

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. The first author thanks department of Mathematics at university of Alberta, and, specifically, professor Anthony T-M Lau for constructive comments and kind support during my sabbatical that led to significant improvements in this study.

References

  1. 1.
    Armendariz, E.P.: A note on extensions of Baer and p.p.-rings. J. Aust. Math. Soc. 18, 470–473 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bell, H.E.: Near-rings in which each element is a power of itself. Bull. Aust. Math. Soc. 2, 363–368 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berberian, S.K.: Baer \(*\)-Rings. Springer, Berlin (1972)CrossRefzbMATHGoogle Scholar
  4. 4.
    Birkenmeier, G.F., Kim, J.Y., Park, J.K.: On quasi-Baer rings. Contemp. Math. 259, 67–92 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Birkenmeier, G.F., Park, J.K.: Triangular matrix representations of ring extensions. J. Algebra 265(2), 457–477 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chatters, A.W., Hajarnavis, C.R.: Rings with Chain Conditions. Pitman Advanced Publishing Program, Boston (1980)zbMATHGoogle Scholar
  7. 7.
    Clark, W.E.: Twisted matrix units semigroup algebras. Duke Math. J. 34, 417–424 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fliess, M.: Sur divers produits de series fonnelles. Bull. Soc. Math. Fr. 102, 181–191 (1974)CrossRefzbMATHGoogle Scholar
  9. 9.
    Habibi, M., Moussavi, A., Manaviyat, R.: On skew quasi-Baer rings. Commun. Algebra 38(10), 3637–3648 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Habibi, M., Moussavi, A.: On nil skew Armendariz rings. Asian Eur. J. Math. 5(2), 1250017 (2012). 16 ppMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hashemi, E., Moussavi, A.: Polynomial extensions of quasi-Baer rings. Acta Math. Hungar. 107(3), 207–224 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hassanein, A.M.: Clean rings of skew Hurwitz series. Le Matematiche 62(1), 47–54 (2007)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hirano, Y.: On ordered monoid rings over a quasi-Baer ring. Commun. Algebra 29, 2089–2095 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hwang, S.U., Jeon, Y.C., Lee, Y.: Structure and topological conditions of NI rings. J. Algebra 302, 186–199 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kaplansky, I.: Projections in Banach algebras. Ann. Math. 53, 235–249 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kaplansky, I.: Rings of operators. Benjamin, New York (1965)zbMATHGoogle Scholar
  17. 17.
    Keigher, W.F.: Adjunctions and comonads in differential algebra. Pacific J. Math. 248, 99–112 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Keigher, W.F.: On the ring of Hurwitz series. Commun. Algebra 25(6), 1845–1859 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Keigher, W.F., Pritchard, F.L.: Hurwitz series as formal functions. J. Pure Appl. Algebra 146, 291–304 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Krempa, J.: Some examples of reduced rings. Algebra Colloq. 3(4), 289–300 (1996)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Lam, T.Y.: A First Course in Noncommutative Rings. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  22. 22.
    Lanski, C.: Nil subrings of Goldie rings are nilpotent. Can. J. Math. 21, 904–907 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lenagan, T.H.: Nil ideals in rings with finite Krull dimension. J. Algebra 29, 77–87 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Liu, Z.: Hermite and PS-rings of Hurwitz series. Commun. Algebra 28(1), 299–305 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Marks, G.: On 2-primal Öre extensions. Commun. Algebra 29, 2113–2123 (2001)CrossRefzbMATHGoogle Scholar
  26. 26.
    Paykan, K.: Nilpotent elements of skew Hurwitz series rings. Rendiconti del Circolo Matematico di Palermo Series 2 65(3), 451–458 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Paykan, K., Moussavi, A.: Baer and quasi-Baer skew generalized power series rings. Commun. Algebra 44(4), 1615–1635 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Paykan, K.: Principally quasi-Baer skew Hurwitz series rings. Boll. Unione Mat. Ital. 10(4), 607–616 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Paykan, K.: A study on skew Hurwitz series rings. Ricerche Mat. 66(2), 383–393 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Paykan, K., Moussavi, A.: Semiprimeness, quasi-Baerness and prime radical of skew generalized power series rings. Commun. Algebra 45(6), 2306–2324 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Pollingher, A., Zaks, A.: On Baer and quasi-Baer rings. Duke Math. J. 37, 127–138 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Rickart, C.E.: Banach algebras with an adjoint operation. Ann. Math. 47, 528–550 (1946)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Sherman, S.: The second adjoint of a \(C^{*}\)-algebra. Proc. Int. Congr. Math. Camb. 1, 470 (1950)Google Scholar
  34. 34.
    Takeda, Z.: Conjugate spaces of operator algebras. Proc. Jpn. Acad. 30, 90–95 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Taft, E.T.: Hurwitz invertibility of linearly recursive sequences. Congr. Numer. 73, 37–40 (1990)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Basic Sciences, South Tehran BranchIslamic Azad UniversityTehranIran
  2. 2.Department of Basic Sciences, Garmsar BranchIslamic Azad UniversityGarmsarIran

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