DOI QR코드

DOI QR Code

ON THE NUMBER OF SEMISTAR OPERATIONS OF SOME CLASSES OF PRUFER DOMAINS

  • Mimouni, Abdeslam (Department of Mathematics and Statistics King Fahd University of Petroleum & Minerals)
  • Received : 2018.12.13
  • Accepted : 2019.04.25
  • Published : 2019.11.30

Abstract

The purpose of this paper is to compute the number of semistar operations of certain classes of finite dimensional $Pr{\ddot{u}}fer$ domains. We prove that ${\mid}SStar(R){\mid}={\mid}Star(R){\mid}+{\mid}Spec(R){\mid}+ {\mid}Idem(R){\mid}$ where Idem(R) is the set of all nonzero idempotent prime ideals of R if and only if R is a $Pr{\ddot{u}}fer$ domain with Y -graph spectrum, that is, R is a $Pr{\ddot{u}}fer$ domain with exactly two maximal ideals M and N and $Spec(R)=\{(0){\varsubsetneq}P_1{\varsubsetneq}{\cdots}{\varsubsetneq}P_{n-1}{\varsubsetneq}M,N{\mid}P_{n-1}{\varsubsetneq}N\}$. We also characterize non-local $Pr{\ddot{u}}fer$ domains R such that ${\mid}SStar(R){\mid}=7$, respectively ${\mid}SStar(R){\mid}=14$.

Acknowledgement

Supported by : KFUPM

References

  1. G. W. Chang and M. Fontana, Uppers to zero and semistar operations in polynomial rings, J. Algebra 318 (2007), no. 1, 484-493. https://doi.org/10.1016/j.jalgebra. 2007.06.010 https://doi.org/10.1016/j.jalgebra.2007.06.010
  2. G. W. Chang, M. Fontana, and M. H. Park, Polynomial extensions of semistar operations, J. Algebra 390 (2013), 250-263. https://doi.org/10.1016/j.jalgebra.2013.05.020 https://doi.org/10.1016/j.jalgebra.2013.05.020
  3. D. E. Dobbs and R. Fedder, Conducive integral domains, J. Algebra 86 (1984), no. 2, 494-510. https://doi.org/10.1016/0021-8693(84)90044-9 https://doi.org/10.1016/0021-8693(84)90044-9
  4. J. Elliott, Semistar operations on Dedekind domains, Comm. Algebra 43 (2015), no. 1, 236-248. https://doi.org/10.1080/00927872.2014.897571 https://doi.org/10.1080/00927872.2014.897571
  5. C. A. Finocchiaro, M. Fontana, and D. Spirito, Spectral spaces of semistar operations, J. Pure Appl. Algebra 220 (2016), no. 8, 2897-2913. https://doi.org/10.1016/j.jpaa.2016.01.008 https://doi.org/10.1016/j.jpaa.2016.01.008
  6. C. A. Finocchiaro and D. Spirito, Some topological considerations on semistar operations, J. Algebra 409 (2014), 199-218. https://doi.org/10.1016/j.jalgebra.2014. 04.002 https://doi.org/10.1016/j.jalgebra.2014.04.002
  7. M. Fontana, P. Jara, and E. Santos, Local-global properties for semistar operations, Comm. Algebra 32 (2004), no. 8, 3111-3137. https://doi.org/10.1081/AGB-120039282 https://doi.org/10.1081/AGB-120039282
  8. G. Fusacchia, Injective modules and semistar operations, J. Pure Appl. Algebra 216 (2012), no. 1, 77-90. https://doi.org/10.1016/j.jpaa.2011.05.004 https://doi.org/10.1016/j.jpaa.2011.05.004
  9. R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, Inc., New York, 1972.
  10. F. Halter-Koch, Localizing systems, module systems, and semistar operations, J. Algebra 238 (2001), no. 2, 723-761. https://doi.org/10.1006/jabr.2000.8671 https://doi.org/10.1006/jabr.2000.8671
  11. W. Heinzer, Integral domains in which each non-zero ideal is divisorial, Mathematika 15 (1968), 164-170. https://doi.org/10.1112/S0025579300002527 https://doi.org/10.1112/S0025579300002527
  12. E. Houston, A. Mimouni, and M. H. Park, Integral domains which admit at most two star operations, Comm. Algebra 39 (2011), no. 5, 1907-1921. https://doi.org/10.1080/00927872.2010.480956 https://doi.org/10.1080/00927872.2010.480956
  13. E. Houston, A. Mimouni, and M. H. Park, Noetherian domains which admit only finitely many star operations, J. Algebra 366 (2012), 78-93. https://doi.org/10.1016/j.jalgebra.2012.05.015 https://doi.org/10.1016/j.jalgebra.2012.05.015
  14. E. Houston, A. Mimouni, and M. H. Park, Integrally closed domains with only finitely many star operations, Comm. Algebra 42 (2014), no. 12, 5264-5286. https://doi.org/10.1080/00927872.2013.837477 https://doi.org/10.1080/00927872.2013.837477
  15. E. Houston, A. Mimouni, and M. H. Park, Star operations on overrings of Noetherian domains, J. Pure Appl. Algebra 220 (2016), no. 2, 810-821. https://doi.org/10.1016/j.jpaa.2015.07.018 https://doi.org/10.1016/j.jpaa.2015.07.018
  16. E. Houston, A. Mimouni, and M. H. Park, Star operations on overrings of Prufer domains, Comm. Algebra 45 (2017), no. 8, 3297-3309. https://doi.org/10.1080/00927872.2016.1236199 https://doi.org/10.1080/00927872.2016.1236199
  17. J. A. Huckaba and I. J. Papick, When the dual of an ideal is a ring, Manuscripta Math. 37 (1982), no. 1, 67-85. https://doi.org/10.1007/BF01239947 https://doi.org/10.1007/BF01239947
  18. W. Krull, Idealtheorie, Zweite, erganzte Au age. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 46, Springer-Verlag, Berlin, 1968.
  19. R. Matsuda, Note on the number of semistar-operations, Math. J. Ibaraki Univ. 31 (1999), 47-53. https://doi.org/10.5036/mjiu.31.47 https://doi.org/10.5036/mjiu.31.47
  20. R. Matsuda, A note on the number of semistar-operations. II, Far East J. Math. Sci. (FJMS) 2 (2000), no. 1, 159-172.
  21. R. Matsuda, On the number of semistar-operations, in Proceedings of the 5th Symposium on Algebra, Languages and Computation (Matsue, 2001), 69-73, Shimane Univ., Matsue, 2002.
  22. R. Matsuda, Note on the number of semistar-operations. III, in Commutative rings, 77-81, Nova Sci. Publ., Hauppauge, NY, 2002.
  23. R. Matsuda, Note on the number of semistar-operations. IV, Sci. Math. Jpn. 55 (2002), no. 2, 345-347.
  24. R. Matsuda, Note on the number of semistar-operations. V, Sci. Math. Jpn. 57 (2003), no. 1, 57-62.
  25. R. Matsuda, Note on the number of semistar operations. VIII, Math. J. Ibaraki Univ. 37 (2005), 53-79. https://doi.org/10.5036/mjiu.37.53 https://doi.org/10.5036/mjiu.37.53
  26. R. Matsuda, Note on the number of semistar operations. XIII, Adv. Algebra Anal. 1 (2006), no. 3, 147-158.
  27. R. Matsuda, Note on the number of semistar operations. X, Math. J. Ibaraki Univ. 38 (2006), 1-19. https://doi.org/10.5036/mjiu.38.1 https://doi.org/10.5036/mjiu.38.1
  28. R. Matsuda, Note on the number of semistar operations. VI, in Focus on commutative rings research, 187-192, Nova Sci. Publ., New York, 2006.
  29. R. Matsuda, Note on the number of semistar operations. XI, Math. J. Ibaraki Univ. 39 (2007), 11-22. https://doi.org/10.5036/mjiu.39.11 https://doi.org/10.5036/mjiu.39.11
  30. R. Matsuda, Note on the number of semistar operations. XIV, Math. J. Ibaraki Univ. 40 (2008), 11-17. https://doi.org/10.5036/mjiu.40.11 https://doi.org/10.5036/mjiu.40.11
  31. R. Matsuda, The semistar operations on certain Prufer domain, Math. J. Ibaraki Univ. 43 (2011), 1-12. https://doi.org/10.5036/mjiu.43.1 https://doi.org/10.5036/mjiu.43.1
  32. R. Matsuda, The semistar operations on certain Prufer domain, II, Math. J. Ibaraki Univ. 46 (2014), 1-8. https://doi.org/10.5036/mjiu.46. https://doi.org/10.5036/mjiu.46.
  33. R. Matsuda, The construction of all the star operations and all the semistar operations on 1-dimensional Prufer domains, Math. J. Ibaraki Univ. 47 (2015), 19-37. https://doi.org/10.5036/mjiu.47.19 https://doi.org/10.5036/mjiu.47.19
  34. R. Matsuda and T. Sugatani, Semistar-operations on integral domains. II, Math. J. Toyama Univ. 18 (1995), 155-161.
  35. A. Mimouni, Krull dimension, overrings and semistar operations of an integral domain, J. Algebra 321 (2009), no. 5, 1497-1509. https://doi.org/10.1016/j.jalgebra.2008.11.028 https://doi.org/10.1016/j.jalgebra.2008.11.028
  36. A. Mimouni and M. Samman, On the cardinality of semistar operations on integral domains, Comm. Algebra 33 (2005), no. 9, 3311-3321. https://doi.org/10.1081/AGB-200058203 https://doi.org/10.1081/AGB-200058203
  37. A. Okabe and R. Matsuda, Semistar-operations on integral domains, Math. J. Toyama Univ. 17 (1994), 1-21.
  38. D. Spirito, Towards a classication of stable semistar operations on a Prufer domain, Comm. Algebra 46 (2018), no. 4, 1831-1842. https://doi.org/10.1080/00927872.2017.1360329 https://doi.org/10.1080/00927872.2017.1360329