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Further results on skew Hurwitz series ring (I)

  • Kamal PaykanEmail author
Article

Abstract

In this paper, we continue the study of skew Hurwitz series ring \((H R, \alpha )\), where R is a ring equipped with an endomorphism \(\alpha \). In particular, we investigate the problem when a skew Hurwitz series series ring \((HR, \alpha )\) has the same Goldie rank as the ring R, and we obtain partial characterizations for it to be serial semiprime. Finally, we will obtain criterion for skew Hurwitz series rings to be right non-singular.

Keywords

Skew Hurwitz series ring Goldie rank Serial semiprime ring Non-singular 

Mathematics Subject Classification

16S99 16W60 16S36 16N40 

Notes

Acknowledgements

The author 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.

References

  1. 1.
    Bell, A.D., Goodearl, K.R.: Uniform rank over differential operator rings and Poincaré–Birkhoff–Witt extensions. Pac. J. Math. 131, 13–37 (1988)CrossRefGoogle Scholar
  2. 2.
    Benhissi, A., Koja, F.: Basic properties of Hurwitz series rings. Ric. Mat. 61(2), 255–273 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fliess, M.: Sur divers produits de series fonnelles. Bull. Soc. Math. Fr. 102, 181–191 (1974)CrossRefGoogle Scholar
  4. 4.
    Goodearl, K.R., Letzter, E.S.: Prime factor algebras of the coordinate ring of quantum matrices. Proc. Am. Math. Soc. 121, 1017–1025 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Grzeszczuk, P.: Goldie dimension of differential operator rings. Commun. Algebra 16, 689–701 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Jategaonkar, A.: Skew polynomial rings over orders in Artinian rings. J. Algebra 21, 51–59 (1972)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Keigher, W.F.: Adjunctions and comonads in differential algebra. Pac. J. Math. 248, 99–112 (1975)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Keigher, W.F.: On the ring of Hurwitz series. Commun. Algebra 25(6), 1845–1859 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Keigher, W.F., Pritchard, F.L.: Hurwitz series as formal functions. J. Pure Appl. Algebra 146, 291–304 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kerr, J.: The power series ring over an Öre domain need not be Öre. J. Algebra 75, 175–177 (1982)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lam, T.Y.: A First Course in Noncommutative Rings. Springer, New York (1991)CrossRefGoogle Scholar
  12. 12.
    Lam, T.Y.: Lectures on Modules and Rings, Graduate Texts in Mathematics, vol. 189. Springer, New York (1999)CrossRefGoogle Scholar
  13. 13.
    Leroy, A., Matczuk, J.: Goldie conditions for ore extensions over semiprime rings. Algebra Represent. Theory 8, 679–688 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Letzter, E.S., Wang, L.: Goldie ranks of skew power series rings of automorphic type. Commun. Algebra 40(6), 1911–1917 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Liu, Z.: Hermite and PS-rings of Hurwitz series. Commun. Algebra 28(1), 299–305 (2000)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Matczuk, J.: Goldie rank of Öre extensions. Commun. Algebra 23, 1455–1471 (1995)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mcconnell, J.C., Robson, J.C.: Non-commutative Noetherian rings. Wiley, Chichester (1987)zbMATHGoogle Scholar
  18. 18.
    Paykan, K.: Nilpotent elements of skew Hurwitz series rings. Rend. del Circ. Mat. di Palermo Ser. 2 65(3), 451–458 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Paykan, K., Moussavi, A.: Study of skew inverse Laurent series rings. J. Algebra Appl. 16(11), 1750221 (2017). (33 pages)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Paykan, K.: Principally quasi-Baer skew Hurwitz series rings. Boll. Unione Mat. Ital. 10(4), 607–616 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Paykan, K.: A study on skew Hurwitz series rings. Ric. mat. 66(2), 383–393 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Paykan, K., Daneshmand, H.: Zero divisor graphs of skew Hurwitz series rings. Le Mat. 73(1), 25–40 (2018)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Paykan, K.: Goldie Ranks of skew generalized power series rings (submitted) Google Scholar
  24. 24.
    Puninski, G.: Serial Rings. Kluwer Academic Publishers, Dordrecht (2001)CrossRefGoogle Scholar
  25. 25.
    Shock, R.C.: Polynomial rings over finite dimension rings. Pac. J. Math. 42, 251–257 (1972)CrossRefGoogle Scholar
  26. 26.
    Taft, E.T.: Hurwitz invertibility of linearly recursive sequences. Congr. Numer. 73, 37–40 (1990)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Tehranchi, A., Paykan, K.: Some results on skew Hurwitz series rings. Rend. del Circ. Mat. di Palermo Ser. 2 68(2), 329–337 (2019)zbMATHGoogle Scholar
  28. 28.
    Tuganbaev, A.A.: Polynomial and series rings and principal ideals. J. Math. Sci. 114, 1204–1226 (2003)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Warfield, R.B.: Serial rings and finitely presented modules. J. Algebra 37, 187–222 (1975)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Warfield, R.B.: Prime ideals in ring extensions. J. Lond. Math. Soc. (2) 28, 453–460 (1983)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Basic Sciences, Garmsar BranchIslamic Azad UniversityGarmsarIran

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