On \(\star \)-Semi-homogeneous Integral Domains

  • D. D. AndersonEmail author
  • Muhammad Zafrullah
Part of the Trends in Mathematics book series (TM)


Let \(\star \) be a finite character star-operation defined on an integral domain D. A nonzero finitely generated ideal of D is \(\star \)-homogeneous if it is contained in a unique maximal \(\star \)-ideal. And D is called a \(\star \)-semi-homogeneous (\(\star \)-SH) domain if every proper nonzero principal ideal of D is a \(\star \)-product of \(\star \)-homogeneous ideals. Then D is a \(\star \)-semi-homogeneous domain if and only if the intersection D \(=\) \(\underset{P\in \star \text {-}{\text {Max}}(D)}{\bigcap D_{P}}\) is independent and locally finite where \(\star \)-\({\text {Max}}(D)\) is the set of maximal \(\star \)-ideals of D. The \(\star \)-SH domains include h-local domains, weakly Krull domains, Krull domains, generalized Krull domains, and independent rings of Krull type. We show that by modifying the definition of a \(\star \)-homogeneous ideal we get a theory of each of these special cases of \(\star \)-SH domains.


  1. 1.
    D.D. Anderson, \(\pi \)-domains, overrings, and divisorial ideals. Glasgow Math. J. 19, 199–203 (1978)MathSciNetCrossRefGoogle Scholar
  2. 2.
    D.D. Anderson, Star-operations induced by overrings. Commun. Algebra 16, 2535–2553 (1988)MathSciNetCrossRefGoogle Scholar
  3. 3.
    D.D. Anderson, D.F. Anderson, Generalized GCD domains. Comment. Math. Univ. St. Pauli. 27, 215–221 (1979)Google Scholar
  4. 4.
    D.D. Anderson, S.J. Cook, Two star operations and their induced lattices. Commun. Algebra. 28, 2461–2476 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    D.D. Anderson, E.G. Houston, M. Zafrullah, t-linked extensions, the t-class group, and Nagata’s theorem. J. Pure Appl. Algebra 86, 109–124 (1993)MathSciNetCrossRefGoogle Scholar
  6. 6.
    D.D. Anderson, L.A. Mahaney, On primary factorization. J. Pure Appl. Algebra 54, 141–154 (1988)MathSciNetCrossRefGoogle Scholar
  7. 7.
    D.D. Anderson, J.L. Mott, M. Zafrullah, Finite character representations of integral domains. Boll. Un. Mat. Ital. 6(B(7)), 613–630 (1992)Google Scholar
  8. 8.
    D.D. Anderson, M. Zafrullah, Weakly factorial domains and groups of divisibility. Proc. Am. Math. Soc. 109, 907–913 (1990)MathSciNetCrossRefGoogle Scholar
  9. 9.
    D.D. Anderson, M. Zafrullah, Almost Bezout domains. J. Algebra 142, 285–309 (1991)MathSciNetCrossRefGoogle Scholar
  10. 10.
    D.D. Anderson, M. Zafrullah, Independent locally-finite intersections of localizations. Houston J. Math. 25, 109–124 (1999)Google Scholar
  11. 11.
    D.F. Anderson, A general theory of class groups. Commun. Algebra 16, 805–847 (1988)MathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Bouvier, Le groupe de classes d’un anneau integre. 107 erne Congres National des Societe Savantes. Brest, France, Fasc. IV, 85–92 (1982)Google Scholar
  13. 13.
    A. Bouvier, M. Zafrullah, On some class groups of an integral domain. Bull. Soc. Math. Greece 29, 45–49 (1988)Google Scholar
  14. 14.
    R. Gilmer, Multiplicative Ideal Theory (Marcel Dekker, New York, 1972)Google Scholar
  15. 15.
    M. Griffin, Rings of Krull type. J. Reine Angew. Math. 229, 1–27 (1968)Google Scholar
  16. 16.
    F. Halter-Koch, Ideal Systems - An Introduction to Multiplicative Ideal Theory (Marcel Dekker, New York, 1998)Google Scholar
  17. 17.
    E.G. Houston, M. Zafrullah, \(\star \) Super potent domains. J. Commut. Algebra, to appearGoogle Scholar
  18. 18.
    P. Jaffard, Les Systèmes d’ Ideáux (Dunod, Paris, 1962)Google Scholar
  19. 19.
    I. Kaplansky, Commutative Rings, revised edn (Polygonal Publishing, Washington, 1994)Google Scholar
  20. 20.
    E. Matlis, Torsion-Free Modules (The University of Chicago Press, Chicago, 1972)Google Scholar
  21. 21.
    U. Storch, Fastfaktorielle Ringe. Schritenreiche Math. Inst. Univ. Munster, vol. 36, Munster (1967)Google Scholar
  22. 22.
    M. Zafrullah, A general theory of almost factoriality. Manuscripta Math. 51, 29–62 (1985)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe University of IowaIowa CityUSA
  2. 2.Department of MathematicsIdaho State UniversityPocatelloUSA

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