Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

LCM-splitting sets in some ring extensions


Authors: Tiberiu Dumitrescu and Muhammad Zafrullah
Journal: Proc. Amer. Math. Soc. 130 (2002), 1639-1644
MSC (2000): Primary 13A05, 13A15; Secondary 13B02, 13B22
DOI: https://doi.org/10.1090/S0002-9939-01-06301-8
Published electronically: November 15, 2001
MathSciNet review: 1887010
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $S$ be a saturated multiplicative set of an integral domain $D$. Call $S$an lcm splitting set if $dD_{S}\cap D$ and $dD\cap sD$ are principal ideals for every $d\in D$ and $s\in S$. We show that if $R$ is an $R_{2}$-stable overring of $D$ (that is, if whenever $a,b\in D$ and $aD\cap bD$ is principal, it follows that $(aD\cap bD)R=aR\cap bR)$ and if $S$ is an lcm splitting set of $D$, then the saturation of $S$ in $R$ is an lcm splitting set in $R$. Consequently, if $D$ is Noetherian and $p\in D$ is a (nonzero) prime element, then $p$ is also a prime element of the integral closure of $ D $. Also, if $D$ is Noetherian, $S$ is generated by prime elements of $D$and if the integral closure of $D_{S}$ is a UFD, then so is the integral closure of $D$.


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

  • [AAZ] D.D. ANDERSON, D.F. ANDERSON, M. ZAFRULLAH, Splitting the t-class group, J. Pure Appl. Algebra 74(1991), 17-37. MR 93d:13023
  • [AAZ1] D.D. ANDERSON, D.F. ANDERSON, M. ZAFRULLAH, Factorization in integral domains II, J. Algebra 152(1992), 78-93. MR 94c:13019
  • [AZ] D. D. ANDERSON, M. ZAFRULLAH, P.M. Cohn's completely primal elements, Lecture Notes in Pure and Applied Mathematics, vol. 171, Marcel Dekker, New York, 1995, pp. 115-123. MR 96g:13024
  • [AZ1] D. D. ANDERSON, M. ZAFRULLAH, Splitting sets in integral domains, Proc. Amer. Math Soc. 129(2001), 2209-2217. CMP 2001:11
  • [ADR] D.F. ANDERSON, D.E. DOBBS, M. ROITMAN, Root closure in commutative rings, Ann. Sci. Univ. Clermont-Ferrand II Math. 26(1990), 1-11. MR 92f:13006
  • [AHZ] D.F. ANDERSON, E. HOUSTON, M. ZAFRULLAH, Pseudo-integrality, Canad. Math. Bull. 34(1991), 15-22. MR 92f:13025
  • [B] N. BOURBAKI, ``Algèbres Commutative,'' Hermann, Paris, 1965. MR 41:5339
  • [CMZ] D. COSTA, J. MOTT, M. ZAFRULLAH, The construction $D+XD_{S}[X]$, J. Algebra. 53(1978), 423-439. MR 58:22046
  • [DHLZ] D. DOBBS, E. HOUSTON, T. LUCAS, M. ZAFRULLAH, t-linked overrings and Prüfer v-multiplication domains, Comm. Algebra, 17(1989), 2835-2852. MR 90j:13016
  • [DHLRZ] D. DOBBS, E. HOUSTON, T. LUCAS, M. ROITMAN, M. ZAFRULLAH, On t-linked overrings, Comm. Algebra, 20(1992), 1463-1488. MR 93e:13034
  • [F] R. FOSSUM, ``The Divisor Class Group of a Krull Domain,'' Springer, New York, 1973. MR 52:3139
  • [G] R. GILMER, ``Multiplicative Ideal Theory,'' Marcel Dekker, New York, 1972. MR 55:323
  • [MS] J.L. MOTT, M. SCHEXNAYDER, Exact sequences of semi-value groups, J. Reine Angew. Math. 283/284(1976), 388-401. MR 53:8050
  • [GP] R. GILMER, T. PARKER, Divisibility properties in semigroup rings, Michigan J. Math. 21(1974), 65-86. MR 49:7381
  • [S] U. STORCH, Fastfaktorielle Ringe, Schriftenreihe Math. Ins. Univ. Münster 36(1967). MR 35:5437
  • [U] H. UDA, LCM-stableness in ring extensions, Hiroshima Math. J. 13(1983), 357-377. MR 85b:13008
  • [Z] M. ZAFRULLAH, A general theory of almost factoriality, Manuscripta Math. 51(1985), 29-62. MR 86m:13023

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13A05, 13A15, 13B02, 13B22

Retrieve articles in all journals with MSC (2000): 13A05, 13A15, 13B02, 13B22


Additional Information

Tiberiu Dumitrescu
Affiliation: Facultatea de Matematică, Universitatea Bucureşti, Str. Academiei 14, Bucharest, RO-70190, Romania
Email: tiberiu@al.math.unibuc.ro

Muhammad Zafrullah
Affiliation: Department of Mathematics, Idaho State University, Pocatello, Idaho 83209-8085
Email: zufrmuha@isu.edu

DOI: https://doi.org/10.1090/S0002-9939-01-06301-8
Keywords: lcm-splitting set, $R_{2}$-stable overring, Noetherian domain
Received by editor(s): May 24, 2000
Received by editor(s) in revised form: January 15, 2001
Published electronically: November 15, 2001
Additional Notes: The authors gratefully acknowledge the referee’s interest in improving the presentation of this paper.
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society