LOCALLY DIVIDED DOMAINS OF THE FORM $D[X]_N_v$

• Chang, Gyu Whan (Department of Mathematics University of Incheon)
• Accepted : 2010.03.05
• Published : 2010.03.01

Abstract

Let D be an integral domain, X be an indeterminate over D, and $N_v=\{f{\in}D[X]{\mid}(A_f)_v=D\}$. In this paper, we introduce the concept of t-locally divided domains, and we then prove that $D[X]_{N_v}$ is a locally divided domain if and only if D is a t-locally divided UMT-domain, if and only if D[X] is a t-locally divided domain.

Acknowledgement

Supported by : University of Incheon

References

1. A. Badawi, On divided commutative rings, Comm. Algebra 27(1999), 1465-1474. https://doi.org/10.1080/00927879908826507
2. P.-J. Cahen, J.-L. Chabert, D.E. Dobbs, and F. Tartarone, On locally divided domains of the form Int(D), Arch. Math. 74(2000), 183-191. https://doi.org/10.1007/s000130050429
3. G.W. Chang, Strong Mori domains and the ring $D[X]_{N_v}$ , J. Pure Appl. Algebra 197(2005), 293-304. https://doi.org/10.1016/j.jpaa.2004.08.036
4. G.W. Chang, Locally pseudo-valuation domains of the form $D[X]_{N_v}$ , J. Korean Math. Soc. 45(2008), 1405-1416. https://doi.org/10.4134/JKMS.2008.45.5.1405
5. G.W. Chang and M. Zafrullah, The w-integral closure of integral domains, J. Algebra 259(2006), 195-210.
6. D.E. Dobbs, Divided rings and going-down, Pacific J. Math. 67(1976), 353-363. https://doi.org/10.2140/pjm.1976.67.353
7. D.E. Dobbs, On locally divided integral domains and CPI-overrings, Internat. J. Math. Sci. 4(1981), 119-135. https://doi.org/10.1155/S0161171281000082
8. D.E. Dobbs and M. Fontana, Locally pseudo-valuation domains, Ann. Mat. Pura Appl.(4), 134(1983), 147-168. https://doi.org/10.1007/BF01773503
9. M. Fontana, S. Gabelli, and E. Houston, UMT-domains and domains with Prufer integral closure, Comm. Algebra 26 (1998), 1017-1039. https://doi.org/10.1080/00927879808826181
10. R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972.
11. J.R. Hedstrom and E.G. Houston, Pseudo-valuation domains, Pacific J. Math.75(1978), 137-147. https://doi.org/10.2140/pjm.1978.75.137
12. E. Houston and M. Zafrullah, On t-invertibility, II, Comm. Algebra 17(1989), 1955-1969. https://doi.org/10.1080/00927878908823829
13. B.G. Kang, Prufer v-multiplication domains and the ring $R[X]_{N_v}$ , J. Algebra 123 (1989), 151-170. https://doi.org/10.1016/0021-8693(89)90040-9