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Coherence in Bi-amalgamated Algebras Along Ideals

  • Mounir El Ouarrachi
  • Najib MahdouEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 228)

Abstract

Let \(f: A\longrightarrow B\) and \(g: A\longrightarrow C\) be two ring homomorphisms and let J (resp., \(J'\)) be an ideal of B (resp., C) such that \(f^{-1}(J)=g^{-1}(J')\). In this paper, we investigate the transfer of the property of coherence in the bi-amalgamation of A with (BC) along \((J,J')\) with respect to (fg) (denoted by \(A\bowtie ^{f,g}(J,J'))\), introduced and studied by Kabbaj, Louartiti, and Tamekkante in 2013. We provide necessary and sufficient conditions for \(A\bowtie ^{f,g}(J,J')\) to be a coherent ring.

Keywords

Bi-amalgamated algebra Amalgamated algebra Coherence 

Notes

Acknowledgment

The authors would like to express their sincere thanks to the referee for his/her helpful suggestions and comments.

References

  1. 1.
    Aloui Ismaili, K., Mahdou, N.: Coherence in amalgamated algebra along an ideal. Bull Iranian Math. Soc. 41(3), 1–9 (2015)Google Scholar
  2. 2.
    Barucci, V., Anderson, D.F., Dobbs, D.E.: Coherent Mori domains and the principal ideal theorem. Comm. Algebra 15, 1119–1156 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boisen, M.B., Sheldon, P.B.: CPI-extension: Over rings of integral domains with special prime spectrum. Canad. J. Math. 29, 722–737 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brewer, W., Rutter, E.: \(D+M\) constructions with general overrings. Michigan Math. J. 23, 33–42 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chhiti, M., Jarrar, M., Kabbaj, S., Mahdou, N.: Prüfer conditions in an amalgamated duplication of a ring along an ideal. Commun. Algebra 43(1), 249–261 (2015)Google Scholar
  6. 6.
    Costa, D.: Parameterizing families of non-Noetherian rings. Commun. Algebra 22, 3997–4011 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    D’Anna, M.: A construction of Gorenstein rings. J. Algebra 306(2), 507–519 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    D’Anna, M., Fontana, M.: amalgamated duplication of a ring along a multiplicative-canonical ideal. Ark. Mat. 45(2), 241–252 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D’Anna, M., Fontana, M.: An amalgamated duplication of a ring along an ideal: the basic properties. J. Algebra Appl. 6(3), 443–459 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D’Anna, M., Finocchiaro, C.A., Fontana, M.: Amalgamated algebras along an ideal. In: Commutative Algebra and Applications, Proceedings of the Fifth International Fez Conference on Commutative Algebra and Applications, Fez, Morocco. W. de Gruyter Publisher, Berlin, pp. 155–172 (2009)Google Scholar
  11. 11.
    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
  12. 12.
    Dobbs, D.E., Kabbaj, S., Mahdou, N., Sobrani, M.: When is \(D+M\) \(n\)-coherent and an \((n,d)\)-domain. Lecture Notes in Pure and Applications and Mathematics, Dekker, vol. 205, pp. 257–270 (1999)Google Scholar
  13. 13.
    Dobbs, D.E., Papick, I.: When is \(D+M\) coherent. Proc. Am. Math. Soc. 56, 51–54 (1976)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Gabelli, S., Houston, E.: Coherent like conditions in pullbacks. Michigan Math. J. 44, 99–123 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Glaz, S.: Commutative Coherent Rings. Lecture Notes in Mathematics, vol. 1371. Springer, Berlin (1989)Google Scholar
  16. 16.
    Glaz, S.: Finite conductor rings. Proc. Am. Math. Soc. 129, 2833–2843 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Glaz, S.: Controlling the Zero-Divisors of a Commutative Ring. Lecture Notes in Pure and Applications and Mathematics, vol. 231, pp. 191–212. Dekker (2003)Google Scholar
  18. 18.
    Huckaba, J.A.: Commutative Rings with Zero Divisors. Marcel Dekker, New York-Basel (1988)zbMATHGoogle Scholar
  19. 19.
    Kabbaj, S., Louartiti, K., Tamekkente, M.: Bi-amalgmeted algebras along ideals. J. Commun. Algebra 9(1), 65–87 (2017)CrossRefzbMATHGoogle Scholar
  20. 20.
    Kabbaj, S., Mahdou, N.: Trivial extensions defined by coherent-like conditions. Commun. Algebra 32(10), 3937–3953 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nagata, M.: Local Rings. Interscience, New York (1962)zbMATHGoogle Scholar
  22. 22.
    Rotman, J.J.: An Introduction to Homological Algebra. Academic Press, New York (1979)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and Technology of FezUniversity S. M. Ben Abdellah FezFesMorocco

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