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Proceedings of the American Mathematical Society

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

 
 

 

Integral domains with finitely generated groups of divisibility


Author: D. D. Anderson
Journal: Proc. Amer. Math. Soc. 112 (1991), 613-618
MSC: Primary 13A05; Secondary 13G05
DOI: https://doi.org/10.1090/S0002-9939-1991-1055765-X
MathSciNet review: 1055765
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Abstract: Let $ D$ be an integral domain with integral closure $ \overline D $. We show that the group of divisibility $ G(D)$ of $ D$ is finitely generated if and only if $ G(\overline D )$ is finitely generated and $ \overline D /[D:\overline D ]$ is finite. We also show that $ G(D)$ is finitely generated if and only if the monoid of finitely generated fractional ideals of $ D$ (under multiplication) is finitely generated.


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

  • [1] D. D. Anderson and J. L. Mott, Cohen-Kaplansky domains: integral domains with a finite number of irreducible elements, J. Algebra (to appear). MR 1161563 (93e:13041)
  • [2] B. Glastadand J. L. Mott, Finitely generated groups of divisibility, Contemp. Math. 8 (1982), 231-247. MR 653184 (83h:13001)
  • [3] I. Kaplansky, Commutative rings, rev. ed., Univ. of Chicago Press, Chicago, 1974. MR 0345945 (49:10674)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1055765-X
Article copyright: © Copyright 1991 American Mathematical Society

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