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

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

Abstract

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.

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Copyright information

© 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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