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Prüfer property in amalgamated algebras along an ideal

  • Najib Mahdou
  • Moutu Abdou Salam MoutuiEmail author
Article
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Abstract

Let \(f : A \rightarrow B\) be a ring homomorphism and J be an ideal of B. In this paper, we give a characterization of zero divisors of the amalgamation which is a generalization of Maimani’s and Yassemi’s work (see Maimani and Yassemi in J Pure Appl Algebra 212(1):168–174, 2008). Furthermore, we investigate the transfer of Prüfer domain concept to commutative rings with zero divisors in the amalgamation of A with B along J with respect to f (denoted by \(A\bowtie ^fJ),\) introduced and studied by D’Anna et al. (Commutative algebra and its applications, Walter de Gruyter, Berlin, 2009, J Pure Appl Algebra 214:1633–1641, 2010). Our results recover well known results on duplications. The main applications constist in the construction of new original classes of Prüfer rings that are not Gaussian and Prüfer rings with weak global dimension strictly greater than 1.

Keywords

Amalgamated algebra along an ideal Prüfer rings Gaussian rings Amalgamated duplication Trivial rings extension 

Mathematics Subject Classification

13F05 13A15 13E05 13F20 13F30 13D05 16D40 

Notes

References

  1. 1.
    Abuihlail, J., Jarrar, M., Kabbaj, S.: Commutative rings in which every finitely generated ideal is quasiprojective. J. Pure Appl. Algebra 215, 2504–2511 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bakkari, C., Kabbaj, S., Mahdou, N.: Trivial extensions defined by \(Pr\ddot{u}fer\) conditions. J. Pure Appl. Algebra 214, 53–60 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bazzoni, S., Glaz, S.: Prüfer rings, Multiplicative Ideal Theory in Commutative Algebra, Springer, New York, 55–72 (2006)Google Scholar
  4. 4.
    Bazzoni, S., Glaz, S.: Gaussian properties of total rings of quotients. J. Algebra 310, 180–193 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boisen, M., Sheldon, P.B.: CPI-extension: over rings of integral domains with special prime spectrum. Can. J. Math. 29, 722–737 (1977)CrossRefzbMATHGoogle Scholar
  6. 6.
    Butts, H.S., Smith, W.: Prüfer rings. Math. Z. 95, 196–211 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chhiti, M., Jarrar, M., Kabbaj, S., Mahdou, N.: \(Pr\ddot{u}fer\) conditions in an amalgamated duplication of a ring along an ideal. Commun. Algebra 43(1), 249–261 (2015)CrossRefzbMATHGoogle Scholar
  8. 8.
    D’Anna, M., Finocchiaro, C.A., Fontana, M.: Amalgamated algebras along an ideal, Commutative algebra and its applications, Walter de Gruyter, Berlin, pp. 241–252 (2009)Google Scholar
  9. 9.
    D’Anna, M., Finocchiaro, C.A., Fontana, M.: Properties of chains of prime ideals in amalgamated algebras along an ideal. J. Pure Appl. Algebra 214, 1633–1641 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D’Anna, M., Finocchiaro, C.A., Fontana, M.: Algebraic and topological properties of an amalgamated algebra along an ideal. Commun. Algebra 44, 1836–1851 (2016)CrossRefzbMATHGoogle Scholar
  11. 11.
    D’Anna, M., Fontana, M.: The amalgamated duplication of a ring along a multiplicative-canonical ideal. Ark. Mat. 45, 241–252 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    D’Anna, M., Fontana, M.: An amalgamated duplication of a ring along an ideal : the basic properties. J. Algebra Appl. 6, 443–459 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Finocchiaro, C.: Prüfer-like conditions on an amalgamated algebra along an ideal. Houston J. Math. 40(1), 63–79 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fuchs, L.: Uber die Ideale arithmetischer Ringe. Comment. Math. Helv. 23, 334–341 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Glaz, S.: The weak global dimension of Gaussian rings. Proc. Am. Math. Soc. 133(9), 2507–2513 (2005)CrossRefzbMATHGoogle Scholar
  16. 16.
    Glaz, S.: Prüfer conditions in rings with zero-divisors, CRC Press Series of Lectures in Pure Appl. Math. 241, 272–282 (2005)Google Scholar
  17. 17.
    Griffin, M.: Prüfer rings with zero-divisors. J. Reine Angew Math. 239(240), 55–67 (1969)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Jensen, C.U.: Arithmetical rings. Acta Math. Hungr. 17, 115–123 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kabbaj, S., Mahdou, N., Moutui, M.A.S.: Bi-amalgamations subject to the arithmetical property. J. Algebra Appl. 16, 1750030 (11 pages) (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Krull, W.: Breitrage zur arithmetik kommutativer integritatsebereiche. Maths. Z 41, 545–577 (1936)CrossRefzbMATHGoogle Scholar
  21. 21.
    Lucas, T.G.: Gaussian polynomials and invertibility. Proc. Am. Math. Soc. 133(7), 1881–1886 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Mahdou, N., Moutui, M.A.S.: Amalgamated algebras along an ideal defined by Gaussian condition. J. Taibah Univ. Sci. 9(3), 373–379Google Scholar
  23. 23.
    Mahdou, N., Moutui, M. A. S.: fqp-property in amalgamated algebras along an ideal. Asian-Eur. J. Math. 8(3), 1550050 (10 pages) (2015)Google Scholar
  24. 24.
    Mahdou, N., Moutui, M.A.S.: On (A)-rings and strong (A)-rings issued from amalgamations. Stud. Sci. Math. Hung. 55(2), 270–279 (2018)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Maimani, H., Yassemi, S.: Zero-divisor graphs of amalgamated duplication of a ring along an ideal. J. Pure Appl. Algebra 212(1), 168–174 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nagata, M.: Local Rings. Interscience, New York (1962)zbMATHGoogle Scholar
  27. 27.
    Prüfer, H.: Untersuchungen uber teilbarkeitseigenschaften in korpern. J. Reine Angew. Math. 168, 1–36 (1932)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Tsang, H.: Gauss’s Lemma. University of Chicago, Chicago (1965). Ph.D. thesisGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and Technology of FezUniversity S. M. Ben AbdellahFezMorocco
  2. 2.Department of Mathematics, College of ScienceKing Faisal UniversityAl-AhsaaSaudi Arabia

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