Integral domains with finitely generated groups of divisibility
HTML articles powered by AMS MathViewer
- by D. D. Anderson PDF
- Proc. Amer. Math. Soc. 112 (1991), 613-618 Request permission
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
- D. D. Anderson and J. L. Mott, Cohen-Kaplansky domains: integral domains with a finite number of irreducible elements, J. Algebra 148 (1992), no. 1, 17–41. MR 1161563, DOI 10.1016/0021-8693(92)90234-D
- Bruce Glastad and Joe L. Mott, Finitely generated groups of divisibility, Ordered fields and real algebraic geometry (San Francisco, Calif., 1981), Contemp. Math., vol. 8, Amer. Math. Soc., Providence, R.I., 1982, pp. 231–247. MR 653184
- Irving Kaplansky, Commutative rings, Revised edition, University of Chicago Press, Chicago, Ill.-London, 1974. MR 0345945
Additional Information
- © Copyright 1991 American Mathematical Society
- 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