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Some results on t–locally Strong Mori domains and their rings of integer–valued polynomials

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Abstract

An integral domain D is said to be t–locally Strong Mori (for short, t–LSM) if \(D_\mathfrak {m}\) is Strong Mori for all t–maximal ideals \(\mathfrak {m}\) of D. This paper studies some ring–theoretic properties of t–LSM domains and the algebra structure of rings of integer–valued polynomials arising from t–LSM domains. Among other things, we investigate the property of being a t–LSM domain in the t–flat overring extension, the t–Nagata ring, the polynomial ring, pullback construction and the power series ring. Also, we study \({\mathrm {Int}}(D)\) over a t–LSM domain D. Precisely, we are interested in the Krull dimension, the trivial case and some module structure properties.

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References

  1. Anderson, D.D., Anderson, D.F., Markanda, R.: The rings \(R(X)\) and \(R\langle X\rangle\). J. Algebra 95, 96–115 (1985)

    Article  MathSciNet  Google Scholar 

  2. Arnold, J.T.: On the ideal theory of the Kronecker function ring and the domain \(D(X)\). Can. J. Math. 21, 558–563 (1969)

    Article  MathSciNet  Google Scholar 

  3. Barucci, V.: Mori domains. In: Chapman, S., Galz, S. (eds.) Non-Noetherian Commutative Ring Theory, vol. 520, pp. 57–73. Kluwer, London (2000)

    Chapter  Google Scholar 

  4. Cahen, P.-J., Chabert, J.-L.: Integer-valued polynomials. Mathematical Surveys and Monographs. American Mathematical society. 48, 322 (1997)

  5. Cahen, P.-J., Fontana, M., Frisch, S., Glaz, S.: Open problems in commutative ring theory. In: Fontana, M., Frisch, S., Glaz, S. (eds.) Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, pp. 293–305. Springer, Berlin (2014)

    Google Scholar 

  6. Cahen, P.-J., Gabelli, S., Houston, E.: Mori domains of integer-valued polynomials. J. Pure Appl. Algebra 153, 1–15 (2000)

    Article  MathSciNet  Google Scholar 

  7. Cahen, P.-J., Loper, A., Tartarone, F.: Integer-valued polynomials and Prüfer \(v\)-multiplication domains. J. Algebra 226, 765–787 (2000)

    Article  MathSciNet  Google Scholar 

  8. Chabert, J.-L.: Integer-valued polynomials, Prüfer domains and localization. Proc. Am. Math. Soc. 118(4), 1061–1073 (1993)

    MATH  Google Scholar 

  9. Chang, G.W.: Strong Mori domains and the ring \(D[X]_{N_v}\). J. Pure Appl. Algebra 197, 293–304 (2005)

    Article  MathSciNet  Google Scholar 

  10. El Baghdadi, S.: On a class of Prüfer \(v\)-multiplication domains. Commun. Algebra 30, 3723–3742 (2002)

    Article  Google Scholar 

  11. Elliott, J.: Some new approaches to integer-valued polynomial rings. In: Fontana, K., Olberding, S. (eds.) Proceedings of the Fifth International Fez Conference on Commutative Algebra and Applications, pp. 223–237. de Gruyter, New York (2009)

    Google Scholar 

  12. Elliott, J.: Integer-valued polynomial rings, \(t\)-closure, and associated primes. Commun. Algebra 39(11), 4128–4147 (2011)

    Article  MathSciNet  Google Scholar 

  13. Fanggui, W., McCasland, R.L.: On Strong Mori domains. J. Pure Appl. Algebra 135, 155–165 (1999)

    Article  MathSciNet  Google Scholar 

  14. Fontana, M., Izelgue, L., Kabbaj, S., Tartarone, F.: Polynomial closure in essential domains and pullbacks. Pure Appl. Math. 205, 307–321 (1999)

    MathSciNet  MATH  Google Scholar 

  15. Fontana, M., Kabbaj, S.: Essential domains and two conjectures in dimension theory. Proc. Am. Math. Soc. 132, 2529–2535 (2004)

    Article  MathSciNet  Google Scholar 

  16. Gabelli, S., Houston, E.: Coherent-like conditions in pullbacks. Michigan Math. J. 44, 99–122 (1997)

    Article  MathSciNet  Google Scholar 

  17. Gilmer, R.: Multiplicative Ideal Theory. Queen’s Papers in Pure and Applied Mathematics, vol. 90, Queen’s University Kingston, Ontario (1992)

  18. Houston, E., Zafrullah, M.: Integral domains in which each \(t\)-ideal is divisorial. Michigan Math. J. 35, 291–300 (1988)

    Article  MathSciNet  Google Scholar 

  19. Izelgue, L., Mimouni, A., Tamoussit, A.: On the module structure of the integer-valued polynomial rings. Bull. Malays. Math. Sci. Soc. 43, 2687–2699 (2020)

    Article  MathSciNet  Google Scholar 

  20. Kang, B.G.: Prüfer \(v\)-multiplication domains and the ring \(R[X]_{N_v}\). J. Algebra 123, 151–170 (1989)

    Article  MathSciNet  Google Scholar 

  21. Mimouni, A.: TW-domains and Strong Mori domains. J. Pure Appl. Algebra 177, 79–93 (2003)

    Article  MathSciNet  Google Scholar 

  22. Mulay, S.B.: On integer–valued polynomials. In: Zero-Dimensional Commutative Rings, Pure Applied Mathematics, Dekker, New York, vol. 171, pp. 331–345 (1995)

  23. Ouzzaouit, O., Tamoussit, A.: On the transfer of some \(t\)-locally properties. Hacettepe J. Math. Stat. (Accepted)

  24. Park, M.H.: On overrings of Strong Mori domains. J. Pure Appl. Algebra 172(1), 79–85 (2002)

    Article  MathSciNet  Google Scholar 

  25. Park, M.H.: Power series rings over Strong Mori domains. J. Algebra 270(1), 361–368 (2003)

    Article  MathSciNet  Google Scholar 

  26. Querré, J.: Intersections d’anneaux intègres. J. Algebra 43, 55–60 (1976)

    Article  MathSciNet  Google Scholar 

  27. Swanson, I., Huneke, C.: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)

    MATH  Google Scholar 

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Acknowledgements

The author would like to thank the anonymous referee for his/her valuable comments and suggestions that helped to improve the quality of this paper.

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  1. Department of Mathematics, The Regional Center for Education and Training Professions Souss Massa, P.O. Box 106, Inezgane, Morocco

    A. Tamoussit

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  1. A. Tamoussit
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Tamoussit, A. Some results on t–locally Strong Mori domains and their rings of integer–valued polynomials. Rend. Circ. Mat. Palermo, II. Ser 71, 349–360 (2022). https://doi.org/10.1007/s12215-021-00596-9

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