# OVERRINGS OF t-COPRIMELY PACKED DOMAINS

• Kim, Hwan-Koo (DEPARTMENT OF INFORMATION SECURITY HOSEO UNIVERSITY)
• Accepted : 2009.10.14
• Published : 2011.01.01
• 75 6

#### Abstract

It is well known that for a Krull domain R, the divisor class group of R is a torsion group if and only if every subintersection of R is a ring of quotients. Thus a natural question is that under what conditions, for a non-Krull domain R, every (t-)subintersection (resp., t-linked overring) of R is a ring of quotients or every (t-)subintersection (resp., t-linked overring) of R is at. To address this question, we introduce the notions of *-compact packedness and *-coprime packedness of (an ideal of) an integral domain R for a star operation * of finite character, mainly t or w. We also investigate the t-theoretic analogues of related results in the literature.

#### Keywords

t-coprimely packed;t-compactly packed;strong Mori domain;Pr$\ddot{u}$fer v-multiplication domain;tQR-property;(t-)flat

#### Acknowledgement

Supported by : Hoseo University

#### References

1. D. D. Anderson and D. F. Anderson, Locally factorial integral domains, J. Algebra 90 (1984), no. 1, 265-283. https://doi.org/10.1016/0021-8693(84)90214-X
2. G. W. Chang, Strong Mori domains and the ring $D[X]_{N_v}$, J. Pure Appl. Algebra 197 (2005), no. 1-3, 293-304. https://doi.org/10.1016/j.jpaa.2004.08.036
3. G. W. Chang, Prufer *-multiplication domains, Nagata rings, and Kronecker function rings, J. Algebra 319 (2008), no. 1, 309-319. https://doi.org/10.1016/j.jalgebra.2007.10.010
4. G. W. Chang and C. J. Hwang, Covering and intersection conditions for prime ideals, Korean J. Math. 17 (2009), 15-23.
5. D. E. Dobbs, E. G. Houston, T. G. Lucas, M. Roitman, and M. Zafrullah, On t-linked overrings, Comm. Algebra 20 (1992), no. 5, 1463-1488. https://doi.org/10.1080/00927879208824414
6. D. E. Dobbs, E. G. Houston, T. G. Lucas, and M. Zafrullah, t-linked overrings and Prufer v-multiplication domains, Comm. Algebra 17 (1989), no. 11, 2835-2852. https://doi.org/10.1080/00927878908823879
7. D. E. Dobbs, E. G. Houston, T. G. Lucas, and M. Zafrullah, t-linked overrings as intersections of localizations, Proc. Amer. Math. Soc. 109 (1990), no. 3, 637-646.
8. S. El Baghdadi and S. Gabelli, Ring-theoretic properties of PvMDs, Comm. Algebra 35 (2007), no. 5, 1607-1625. https://doi.org/10.1080/00927870601169283
9. V. Erdogdu, Coprimely packed rings, J. Number Theory 28 (1988), no. 1, 1-5. https://doi.org/10.1016/0022-314X(88)90115-1
10. V. Erdogdu, The prime avoidance of maximal ideals in commutative rings, Comm. Algebra 23 (1995), no. 3, 863-868. https://doi.org/10.1080/00927879508825253
11. V. Erdogdu, Three notes on coprime packedness, J. Pure Appl. Algebra 148 (2000), no. 2, 165-170. https://doi.org/10.1016/S0022-4049(00)00003-7
12. V. Erdogdu, Coprime packedness and set theoretic complete intersections of ideals in polynomial rings, Proc. Amer. Math. Soc. 132 (2004), no. 12, 3467-3471. https://doi.org/10.1090/S0002-9939-04-07438-6
13. V. Erdogdu and S. McAdam, Coprimely packed Noetherian polynomial rings, Comm. Algebra 22 (1994), no. 15, 6459-6470. https://doi.org/10.1080/00927879408825200
14. M. Fontana, P. Jara, and E. Santos, Prufer $\star$-multiplication domains and semistar operations, J. Algebra Appl. 2 (2003), no. 1, 21-50. https://doi.org/10.1142/S0219498803000349
15. R. Fossum, The Divisor Class Group of a Krull Domain, Springer-Verlag, New York-Heidelberg, 1973.
16. S. Gabelli, E. G. Houston, and T. G. Lucas, The t#-property for integral domains, J. Pure Appl. Algebra 194 (2004), no. 3, 281-298. https://doi.org/10.1016/j.jpaa.2004.05.002
17. R. Gilmer, Overrings of Prufer domains, J. Algebra 4 (1966), 331-340. https://doi.org/10.1016/0021-8693(66)90025-1
18. R. Gilmer, Multiplicative Ideal Theory, Queen's Papers in Pure and Applied Mathematics, 90, Queen's University, Kingston, Ontario, 1992.
19. R. Gilmer and W. J. Heinzer, Overrings of Prufer domains. II, J. Algebra 7 (1967), 281-302. https://doi.org/10.1016/0021-8693(67)90073-7
20. R. Gilmer and J. Ohm, Integral domains with quotient overrings, Math. Ann. 153 (1964), 97-103. https://doi.org/10.1007/BF01361178
21. E. G. Houston, Prime t-ideals in R[X], Commutative ring theory (Fes, 1992), 163-170, Lecture Notes in Pure and Appl. Math., 153, Dekker, New York, 1994.
22. E. G. Houston and A. Mimouni, On the divisorial spectrum of a Noetherian domain, J. Pure Appl. Algebra 214 (2010), no. 1, 47-52. https://doi.org/10.1016/j.jpaa.2009.04.012
23. B. G. Kang, Prufer v-multiplication domains and the ring $R[X]_{N_v}$, J. Algebra 123 (1989), no. 1, 151-170. https://doi.org/10.1016/0021-8693(89)90040-9
24. I. Kaplansky, Commutative Rings, Revised edition. The University of Chicago Press, Chicago, Ill.-London, 1974.
25. D. J. Kwak and Y. S. Park, On t-flat overrings, Chinese J. Math. 23 (1995), no. 1, 17-24.
26. M. B. Martin and M. Zafrullah, t-linked overrings of Noetherian weakly factorial domains, Proc. Amer. Math. Soc. 115 (1992), 601-604.
27. J. L. Mott, Integral domains with quotient overrings, Math. Ann. 166 (1966), no. 229-232. https://doi.org/10.1007/BF01361169
28. J. L. Mott and M. Zafrullah, On Prufer v-multiplication domains, Manuscripta Math. 35 (1981), no. 1-2, 1-26. https://doi.org/10.1007/BF01168446
29. S. Oda, Radically principal and almost factorial, Bull. Fac. Sci. Ibaraki Univ. Ser. A No. 26 (1994), 17-24.
30. J. V. Pakala and T. S. Shores, On compactly packed rings, Pacific J. Math. 97 (1981), no. 1, 197-201. https://doi.org/10.2140/pjm.1981.97.197
31. N. Popescu, Sur les C. P.-anneaux, C. R. Acad. Sci. Paris Ser. A-B 272 (1971), A1493-A1496.
32. C. M. Reis and T. M. Viswanathan, A compactness property for prime ideals in Noetherian rings, Proc. Amer. Math. Soc. 25 (1970), 353-356. https://doi.org/10.1090/S0002-9939-1970-0254031-6
33. D. E. Rush and L. J. Wallace, Noetherian maximal spectrum and coprimely packed localizations of polynomial rings, Houston J. Math. 28 (2002), no. 3, 437-448.
34. W. Smith, A covering condition for prime ideals, Proc. Amer. Math. Soc. 30 (1971), 451-452. https://doi.org/10.1090/S0002-9939-1971-0282963-2
35. B. Wajnryb and A. Zaks, On the flat overrings of an integral domain, Glasgow Math. J. 12 (1971), 162-165. https://doi.org/10.1017/S0017089500001269
36. 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
37. 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

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