Korean J. Math. Vol. 21 No. 4 (2013) pp.455-462
DOI: https://doi.org/10.11568/kjm.2013.21.4.455

t-splitting sets S of an integral domain D such that DS is a factorial domain

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Published: Dec 11, 2013
Mathematics Subject Classification:
13A15, 13G05

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Gyu Whan Chang
Department of Mathematics, Incheon National University, Incheon 406-772, Republic of Korea

Abstract

Let D be an integral domain, S be a saturated multiplicative subset of D such that DS is a factorial domain, {Xα} be a nonempty set of indeterminates, and D[{Xα}] be the polynomial ring over D. We show that S is a splitting (resp., almost splitting, t-splitting) set in D if and only if every nonzero prime t-ideal of D disjoint from S is principal (resp., contains a primary element, is t-invertible). We use this result to show that D∖{0} is a splitting (resp., almost splitting, t-splitting) set in D[{Xα}] if and only if D is a GCD-domain (resp., UMT-domain with Cl(D[{Xα}]) torsion, UMT-domain).


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References

[1] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Atomic domains in which almost all atoms are prime, Comm. Algebra 20 (1992), 1447–1462. Google Scholar

[2] D.D. Anderson, D.F. Anderson, and M. Zafrullah, Splitting the t-class group, J. Pure Appl. Algebra 74 (1991), 17–37. Google Scholar

[3] D.D. Anderson, D.F. Anderson, and M. Zafrullah, The ring D + XDS[X] and t-splitting sets, Commutative Algebra Arab. J. Sci. Eng. Sect. C Theme Issues 26 (1) (2001), 3–16. Google Scholar

[4] D.D. Anderson, T. Dumitrescu, and M. Zafrullah, Almost splitting sets and AGCD domains, Comm. Algebra 32 (2004), 147–158. Google Scholar

[5] D.F. Anderson and G.W. Chang, Almost splitting sets in integral domains, II, J. Pure Appl. Algebra 208 (2007), 351–359. Google Scholar

[6] A. Bouvier and M. Zafrullah, On some class groups of an integral domain, Bull. Soc. Math. Gr ece (N.S.) 29 (1988), 45-59. Google Scholar

[7] G.W. Chang, Almost splitting sets in integral domains, J. Pure Appl. Algebra 197 (2005), 279–292. Google Scholar

[8] G.W. Chang, Almost splitting sets S of an integral domain D such that DS is a PID, Korean J. Math. 19 (2011), 163–169. Google Scholar

[9] G.W. Chang, T. Dumitrescu, and M. Zafrullah, t-splitting sets in integral do- mains, J. Pure Appl. Algebra 187 (2004), 71–86. Google Scholar

[10] S. El Baghdadi, L. Izelgue, and S. Kabbaj, On the class group of a graded domain, J. Pure Appl. Algebra 171 (2002), 171–184. Google Scholar

[11] M. Fontana, S. Gabelli, and E. Houston, UMT-domains and domains with Pru ̈fer integral closure, Comm. Algebra 26 (1998), 1017–1039. Google Scholar

[12] R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972. Google Scholar

[13] E. Houston and M. Zafrullah, On t-invertibility, II, Comm. Algebra 17 (1989), 1955–1969. Google Scholar

[14] B.G. Kang, Pru ̈fer v-multiplication domains and the ring R[X]Nv , J. Algebra 123 (1989), 151–170. Google Scholar

[15] M. Zafrullah, A general theory of almost factoriality, Manuscripta Math. 51 (1985), 29–62. Google Scholar