Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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PRIME FACTORIZATION OF IDEALS IN COMMUTATIVE RINGS, WITH A FOCUS ON KRULL RINGS

  • Gyu Whan Chang (Department of Mathematics Education Incheon National University) ;
  • Jun Seok Oh (Department of Mathematics Education Jeju National University)
  • Received : 2022.05.30
  • Accepted : 2022.11.08
  • Published : 2023.03.01
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Abstract

Let R be a commutative ring with identity. The structure theorem says that R is a PIR (resp., UFR, general ZPI-ring, π-ring) if and only if R is a finite direct product of PIDs (resp., UFDs, Dedekind domains, π-domains) and special primary rings. All of these four types of integral domains are Krull domains, so motivated by the structure theorem, we study the prime factorization of ideals in a ring that is a finite direct product of Krull domains and special primary rings. Such a ring will be called a general Krull ring. It is known that Krull domains can be characterized by the star operations v or t as follows: An integral domain R is a Krull domain if and only if every nonzero proper principal ideal of R can be written as a finite v- or t-product of prime ideals. However, this is not true for general Krull rings. In this paper, we introduce a new star operation u on R, so that R is a general Krull ring if and only if every proper principal ideal of R can be written as a finite u-product of prime ideals. We also study several ring-theoretic properties of general Krull rings including Kaplansky-type theorem, Mori-Nagata theorem, Nagata rings, and Noetherian property.

Keywords

Acknowledgement

The authors would like to thank the anonymous referee for his/her helpful comments. This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B06029867).

References

  1. J. H. Alajbegovic and E. Osmanagic, Essential valuations of Krull rings with zero divisors, Comm. Algebra 18 (1990), no. 7, 2007-2020. https://doi.org/10.1080/00927879008824008
  2. D. D. Anderson, Multiplication ideals, multiplication rings, and the ring R(X), Canadian J. Math. 28 (1976), no. 4, 760-768. https://doi.org/10.4153/CJM-1976-072-1
  3. D. D. Anderson, Some remarks on the ring R(X), Comment. Math. Univ. St. Paul. 26 (1977/78), no. 2, 137-140. https://doi.org/10.1512/iumj.1977.26.26010
  4. D. D. Anderson, D. F. Anderson, and R. Markanda, The rings R(X) and R⟨X⟩, J. Algebra 95 (1985), no. 1, 96-115. https://doi.org/10.1016/0021-8693(85)90096-1
  5. D. D. Anderson and S. Chun, Some remarks on principal prime ideals, Bull. Aust. Math. Soc. 83 (2011), no. 1, 130-137. https://doi.org/10.1017/S000497271000170X
  6. D. D. Anderson and S. J. Cook, Two star-operations and their induced lattices, Comm. Algebra 28 (2000), no. 5, 2461-2475. https://doi.org/10.1080/00927870008826970
  7. D. D. Anderson and J. A. Huckaba, The integral closure of a Noetherian ring. II, Comm. Algebra 6 (1978), no. 7, 639-657. https://doi.org/10.1080/00927877808822262
  8. D. D. Anderson and R. Markanda, Unique factorization rings with zero divisors, Houston J. Math. 11 (1985), no. 1, 15-30.
  9. D. D. Anderson and R. Markanda, Corrigendum: "Unique factorization rings with zero divisors", Houston J. Math. 11 (1985), no. 3, 423-426.
  10. J. T. Arnold, On the ideal theory of the Kronecker function ring and the domain D(X), Canadian J. Math. 21 (1969), 558-563. https://doi.org/10.4153/CJM-1969-063-4
  11. K. Asano, Uber kommutative Ringe, in denen jedes Ideal als Produkt von Primidealen darstellbar ist, J. Math. Soc. Japan 3 (1951), 82-90. https://doi.org/10.2969/jmsj/00310082
  12. H. S. Butts and R. C. Phillips, Almost multiplication rings, Canadian J. Math. 17 (1965), 267-277. https://doi.org/10.4153/CJM-1965-026-9
  13. G. W. Chang, Weakly factorial rings with zero divisors, in Ideal theoretic methods in commutative algebra (Columbia, MO, 1999), 119-131, Lecture Notes in Pure and Appl. Math., 220, Dekker, New York, 2001.
  14. G. W. Chang, Strong Mori domains and the ring D[X]Nv, J. Pure Appl. Algebra 197 (2005), no. 1-3, 293-304. https://doi.org/10.1016/j.jpaa.2004.08.036
  15. G. W. Chang and B. G. Kang, Integral closure of a ring whose regular ideals are finitely generated, J. Algebra 251 (2002), no. 2, 529-537. https://doi.org/10.1006/jabr.2000.8596
  16. G. W. Chang and B. G. Kang, On Krull rings with zero divisors, Comm. Algebra 49 (2021), no. 1, 207-215. https://doi.org/10.1080/00927872.2020.1797071
  17. G. W. Chang and B. G. Kang, Integral closure of an affine ring, preprint.
  18. E. D. Davis, Overrings of commutative rings. I. Noetherian overrings, Trans. Amer. Math. Soc. 104 (1962), 52-61. https://doi.org/10.2307/1993932
  19. J. Elliott, Rings, modules, and closure operations, Springer Monographs in Mathematics, Springer, Cham, 2019. https://doi.org/10.1007/978-3-030-24401-9
  20. C. R. Fletcher, Unique factorization rings, Proc. Cambridge Philos. Soc. 65 (1969), 579-583. https://doi.org/10.1017/S0305004100003352
  21. C. R. Fletcher, The structure of unique factorization rings, Proc. Cambridge Philos. Soc. 67 (1970), 535-540. https://doi.org/10.1017/s0305004100045825
  22. C. R. Fletcher, Equivalent conditions for unique factorization, Publ. Dep. Math. (Lyon) 8 (1971), no. 1, 13-22.
  23. M. Fontana and K. A. Loper, Nagata rings, Kronecker function rings, and related semistar operations, Comm. Algebra 31 (2003), no. 10, 4775-4805. https://doi.org/10.1081/AGB-120023132
  24. R. M. Fossum, The divisor class group of a Krull domain, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74, Springer-Verlag, New York, 1973.
  25. R. Gilmer, An embedding theorem for HCF-rings, Proc. Cambridge Philos. Soc. 68 (1970), 583-587. https://doi.org/10.1017/s0305004100076568
  26. R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972.
  27. R. Gilmer and W. Heinzer, Primary ideals with finitely generated radical in a commutative ring, Manuscripta Math. 78 (1993), no. 2, 201-221. https://doi.org/10.1007/BF02599309
  28. S. Glaz and W. V. Vasconcelos, Flat ideals. II, Manuscripta Math. 22 (1977), no. 4, 325-341. https://doi.org/10.1007/BF01168220
  29. M. Griffin, Prufer rings with zero divisors, J. Reine Angew. Math. 239(240) (1969), 55-67. https://doi.org/10.1515/crll.1969.239-240.55
  30. M. Griffin, Multiplication rings via their total quotient rings, Canadian J. Math. 26 (1974), 430-449. https://doi.org/10.4153/CJM-1974-043-9
  31. J. R. Hedstrom and E. G. Houston, Some remarks on star-operations, J. Pure Appl. Algebra 18 (1980), no. 1, 37-44. https://doi.org/10.1016/0022-4049(80)90114-0
  32. E. Houston and M. Zafrullah, On t-invertibility. II, Comm. Algebra 17 (1989), no. 8, 1955-1969. https://doi.org/10.1080/00927878908823829
  33. J. A. Huckaba, The integral closure of a Noetherian ring, Trans. Amer. Math. Soc. 220 (1976), 159-166. https://doi.org/10.2307/1997638
  34. J. A. Huckaba, Commutative Rings with Zero Divisors, Decker, New York, 1988.
  35. J. A. Huckaba and I. J. Papick, Quotient rings of polynomial rings, Manuscripta Math. 31 (1980), no. 1-3, 167-196. https://doi.org/10.1007/BF01303273
  36. T. W. Hungerford, On the structure of principal ideal rings, Pacific J. Math. 25 (1968), 543-547. http://projecteuclid.org/euclid.pjm/1102986148 102986148
  37. J. R. Juett, General w-ZPI-rings and a tool for characterizing certain classes of monoid rings, Comm. Algebra, to appear.
  38. J. R. Juett, C. P. Mooney, and L. W. Ndungu, Unique factorization of ideals in commutative rings with zero divisors, Comm. Algebra 49 (2021), no. 5, 2101-2125. https://doi.org/10.1080/00927872.2020.1864390
  39. J. R. Juett, C. P. Mooney, and R. D. Roberts, Unique factorization properties in commutative monoid rings with zero divisors, Semigroup Forum 102 (2021), no. 3, 674-696. https://doi.org/10.1007/s00233-020-10154-x
  40. B. G. Kang, Prufer v-multiplication domains and the ring R[X]Nv, J. Algebra 123 (1989), no. 1, 151-170. https://doi.org/10.1016/0021-8693(89)90040-9
  41. B. G. Kang, On the converse of a well-known fact about Krull domains, J. Algebra 124 (1989), no. 2, 284-299. https://doi.org/10.1016/0021-8693(89)90131-2
  42. B. G. Kang, A characterization of Krull rings with zero divisors, J. Pure Appl. Algebra 72 (1991), no. 1, 33-38. https://doi.org/10.1016/0022-4049(91)90127-N
  43. B. G. Kang, Characterizations of Krull rings with zero divisors, J. Pure Appl. Algebra 146 (2000), no. 3, 283-290. https://doi.org/10.1016/S0022-4049(98)00100-5
  44. I. Kaplansky, Commutative Rings, revised edition, University of Chicago Press, Chicago, IL, 1974.
  45. R. E. Kennedy, A generalization of Krull domains to rings with zero divisors, Thesis (Ph.D), Univ. of Missouri-Kansas, 1973.
  46. R. E. Kennedy, Krull rings, Pacific J. Math. 89 (1980), no. 1, 131-136. http://projecteuclid.org/euclid.pjm/1102779374 102779374
  47. M. Knebusch and T. Kaiser, Manis valuations and Prufer extensions. II, Lecture Notes in Mathematics, 2103, Springer, Cham, 2014. https://doi.org/10.1007/978-3-319-03212-2
  48. W. Krull, Uber die Zerlegung der Hauptideale in allgemeinen Ringen, Math. Ann. 105 (1931), no. 1, 1-14. https://doi.org/10.1007/BF01455805
  49. W. Krull, Allgemeine Bewertungstheorie, J. Reine Angew. Math. 167 (1932), 160-196. http://doi.org/10.1515/crll.1932.167.160
  50. W. Krull, Idealtheorie, Ergebnisse der Math. und ihrer Grenz. vol. 4, No. 3, Berlin, Julius Splinger, 1935.
  51. Max. D. Larsen and P. J. McCarthy, Multiplicative theory of ideals, Pure and Applied Mathematics, Vol. 43, Academic Press, New York, 1971.
  52. K. B. Levitz, A characterization of general Z.P.I.-rings, Proc. Amer. Math. Soc. 32 (1972), 376-380. https://doi.org/10.2307/2037821
  53. T. G. Lucas, Strong Prufer rings and the ring of finite fractions, J. Pure Appl. Algebra 84 (1993), no. 1, 59-71. https://doi.org/10.1016/0022-4049(93)90162-M
  54. M. E. Manis, Valuations on a commutative ring, Proc. Amer. Math. Soc. 20 (1969), 193-198. https://doi.org/10.2307/2035988
  55. J. Marot, Extension de la notion d'anneau valuation, Dept. Math. Faculte dec Sci. de Brest, 46pp. et 39 pp. of complements, 1968.
  56. R. Matsuda, On Kennedy's problems, Comment. Math. Univ. St. Paul. 31 (1982), no. 2, 143-145.
  57. R. Matsuda, Generalizations of multiplicative ideal theory to commutative rings with zerodivisors, Bull. Fac. Sci. Ibaraki Univ. Ser. A 17 (1985), 49-101. https://doi.org/10.5036/bfsiu1968.17.49
  58. S. Mori, Uber die Produktzerlegung der Hauptideale. II, J. Sci. Hiroshima Univ. Ser. A 9 (1939), 145-155. http://doi.org/10.32917/hmj/1558490529
  59. S. Mori, Allgemeine Z.P.I.-Ringe, J. Sci. Hiroshima Univ. Ser. A 10 (1940), 117-136. http://doi.org/10.32917/hmj/1558404030
  60. S. Mori, Uber die Produktzerlegung der Hauptideale. III, J. Sci. Hiroshima Univ. Ser. A 10 (1940), 85-94. http://doi.org/10.32917/hmj/1558404028
  61. Y. Mori, On the integral closure of an integral domain, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 27 (1953), 249-256. https://doi.org/10.1215/kjm/1250777561
  62. J. L. Mott, On invertible ideals in a commutative ring, Louisiana State University, Ph.D. Thesis, 1963.
  63. J. L. Mott, Multiplication rings containing only finitely many minimal prime ideals, J. Sci. Hiroshima Univ. Ser. A-I Math. 33 (1969), 73-83. http://doi.org/10.32917/hmj/1206138588
  64. M. Nagata, Note on integral closures of Noetherian domains, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 28 (1954), 121-124. https://doi.org/10.1215/kjm/1250777425
  65. M. Nagata, On the derived normal rings of Noetherian integral domains, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 29 (1955), 293-303. https://doi.org/10.1215/kjm/1250777189
  66. M. Nagata, Local Rings, Interscience, New York, 1962.
  67. T. Nishimura, Unique factorization of ideals in the sense of quasi-equality, J. Math. Kyoto Univ. 3 (1963), 115-125. https://doi.org/10.1215/kjm/1250524862
  68. T. Nishimura, On the t-ideals of an integral domain, J. Math. Kyoto Univ. 13 (1973), 59-65. https://doi.org/10.1215/kjm/1250523435
  69. A. Okabe and R. Matsuda, Semistar-operations on integral domains, Math. J. Toyama Univ. 17 (1994), 1-21.
  70. D. Portelli and W. Spangher, Krull rings with zero divisors, Comm. Algebra 11 (1983), no. 16, 1817-1851. https://doi.org/10.1080/00927878308822934
  71. U. Storch, Fastfaktorielle Ringe, Schr. Math. Inst. Univ. Munster 36 (1967).
  72. S. J. Tramel, Factorization of principal ideals in the sense of quasi-equality, Doctoral Dissertation, Louisiana State University, 1968.
  73. F. Wang, Modules over R and R{X}, Sichuan Shifan Daxue Xuebao Ziran Kexue Ban 25 (2002), no. 6, 557-562.
  74. F. Wang and H. Kim, Two generalizations of projective modules and their applications, J. Pure Appl. Algebra 219 (2015), no. 6, 2099-2123. https://doi.org/10.1016/j.jpaa.2014.07.025
  75. F. Wang and H. Kim, Foundations of commutative rings and their modules, Algebra and Applications, 22, Springer, Singapore, 2016. https://doi.org/10.1007/978-981-10-3337-7
  76. F. Wang and R. L. McCasland, On w-modules over strong Mori domains, Comm. Algebra 25 (1997), no. 4, 1285-1306. https://doi.org/10.1080/00927879708825920
  77. F. Wang and R. L. McCasland, On strong Mori domains, J. Pure Appl. Algebra 135 (1999), no. 2, 155-165. https://doi.org/10.1016/S0022-4049(97)00150-3
  78. H. Yin, F. Wang, X. Zhu, and Y. Chen, w-modules over commutative rings, J. Korean Math. Soc. 48 (2011), no. 1, 207-222. https://doi.org/10.4134/JKMS.2011.48.1.207
  79. O. Zariski and P. Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, NJ, 1960.
  80. D. Zhou and F. Wang, The strong Mori property in rings with zero divisors, Bull. Korean Math. Soc. 52 (2015), no. 4, 1285-1295. https://doi.org/10.4134/BKMS.2015.52.4.1285