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

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Abstract: Let be an integral domain with integral closure . We show that the group of divisibility of is finitely generated if and only if is finitely generated and is finite. We also show that is finitely generated if and only if the monoid of finitely generated fractional ideals of (under multiplication) is finitely generated.

**[1]**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**, 10.1016/0021-8693(92)90234-D**[2]**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****[3]**Irving Kaplansky,*Commutative rings*, Revised edition, The University of Chicago Press, Chicago, Ill.-London, 1974. MR**0345945**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1991-1055765-X

Article copyright:
© Copyright 1991
American Mathematical Society