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Splitting sets in integral domains


Authors: D. D. Anderson and Muhammad Zafrullah
Journal: Proc. Amer. Math. Soc. 129 (2001), 2209-2217
MSC (1991): Primary 13A05, 13A15, 13G05
DOI: https://doi.org/10.1090/S0002-9939-00-05863-9
Published electronically: December 28, 2000
MathSciNet review: 1823902
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Abstract:

Let $D$ be an integral domain. A saturated multiplicatively closed subset $S$of $D$ is a splitting set if each nonzero $d\in D$ may be written as $d=sa$ where $s\in S$ and $s'D\cap aD=s'aD$ for all $s'\in S$. We show that if $S$ is a splitting set in $D$, then $SU(D_{N})$ is a splitting set in $D_{N}$, $N$ a multiplicatively closed subset of $D$, and that $S\subseteq D$ is a splitting set in $D[X]\iff S$ is an lcm splitting set of $D$, i.e., $S$ is a splitting set of $D$ with the further property that $sD\cap dD$ is principal for all $s\in S$ and $d\in D$. Several new characterizations and applications of splitting sets are given.


References [Enhancements On Off] (What's this?)

  • 1. D.D. Anderson, D.F. Anderson, and M. Zafrullah, Splitting the $t$-class group, J. Pure Appl. Algebra 74 (1991), 17-37. MR 93d:13023
  • 2. D.D. Anderson, D.F. Anderson, and M. Zafrullah, Factorization in integral domains, II, J. Algebra 152 (1992), 78-93. MR 94c:13019
  • 3. D.D. Anderson, J.L. Mott, and M. Zafrullah, Finite character representations for integral domains, Boll. Un. Mat. Ital. B (7) 6 (1992), 613-630. MR 93k:13001
  • 4. D.D. Anderson and M. Zafrullah, Weakly factorial domains and groups of divisibility, Proc. Amer. Math. Soc. 109 (1990), 907-913. MR 90k:13015
  • 5. D.F. Anderson, J. Park, G.-I. Kim, and H.-J. Oh, Splitting multiplicative sets and elasticity, Comm. Algebra 26 (1998), 1257-1276. MR 99c:13002
  • 6. P.M. Cohn, Bezout rings and their subrings, Proc. Cambridge Philos. Soc. 64 (1968), 251-264. MR 36:5117
  • 7. S. Gabelli and M. Roitman, On Nagata's theorem for the class group, J. Pure Appl. Algebra 66 (1990), 31-42. MR 91i:13015
  • 8. R. Gilmer and T. Parker, Divisibility properties in semigroup rings, Michigan Math. J. 21 (1974), 65-86. MR 49:7381
  • 9. J.L. Mott, The group of divisibility of Rees rings, Math. Japon. 20 (1975), 85-87. MR 52:10718
  • 10. J.L. Mott and M. Schexnayder, Exact sequences of semi-value groups, J. Reine Angew. Math. 283/284 (1976), 388-401. MR 53:8050
  • 11. M. Nagata, A remark on the unique factorization theorem, J. Math. Soc. Japan 9 (1957), 143-145. MR 18:869a
  • 12. M. Roitman, On Mori domains and commutative rings with $\mathrm{CC}^{\perp}$ I, J. Pure Appl. Algebra 56 (1989), 247-268. MR 90c:13019
  • 13. M. Zafrullah, On a property of pre-Schreier domains, Comm. Algebra 15 (1987), 1895-1920. MR 88g:13018
  • 14. M. Zafrullah, The $D+XD_{S}[X]$ construction from GCD-domains, J. Pure Appl. Algebra 50 (1988), 93-107. MR 89k:13017

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Additional Information

D. D. Anderson
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242
Email: dan-anderson@uiowa.edu

Muhammad Zafrullah
Affiliation: Department of Mathematics, SCEN 301, The University of Arkansas, Fayetteville, Arkansas 72701
Address at time of publication: Department of Mathematics, Campus Box 8085, Idaho State University, Pocatello, Idaho 83209
Email: kamla@compuserve.com, mzafrullah@usa.net

DOI: https://doi.org/10.1090/S0002-9939-00-05863-9
Keywords: Splitting sets
Received by editor(s): May 5, 1999
Received by editor(s) in revised form: December 18, 1999
Published electronically: December 28, 2000
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2000 American Mathematical Society

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