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
MathSciNet review: 1055765
Full-text PDF Free Access
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.
- 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 https://doi.org/10.1016/0021-8693%2892%2990234-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, The University of Chicago Press, Chicago, Ill.-London, 1974. MR 0345945
D. D. Anderson and J. L. Mott, Cohen-Kaplansky domains: integral domains with a finite number of irreducible elements, J. Algebra (to appear).
B. Glastadand J. L. Mott, Finitely generated groups of divisibility, Contemp. Math. 8 (1982), 231-247.
I. Kaplansky, Commutative rings, rev. ed., Univ. of Chicago Press, Chicago, 1974.