## Dedekind domains and graded rings

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- by Fabien Decruyenaere and Eric Jespers
- Proc. Amer. Math. Soc.
**110**(1990), 21-26 - DOI: https://doi.org/10.1090/S0002-9939-1990-1027092-7
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## Abstract:

We prove that a Dedekind domain $R$, graded by a nontrivial torsionfree abelian group, is either a twisted group ring ${k^t}[G]$ or a polynomial ring $k[X]$, where $k$ is a field and $G$ is an abelian torsionfree rank one group. It follows that $R$ is a Dedekind domain if and only if $R$ is a principal ideal domain. We also investigate the case when $R$ is graded by an arbitrary nontrivial torsionfree monoid.## References

- D. D. Anderson and David F. Anderson,
*Divisibility properties of graded domains*, Canadian J. Math.**34**(1982), no. 1, 196–215. MR**650859**, DOI 10.4153/CJM-1982-013-3 - David F. Anderson,
*Graded Krull domains*, Comm. Algebra**7**(1979), no. 1, 79–106. MR**514866**, DOI 10.1080/00927877908822334 - A. H. Clifford and G. B. Preston,
*The algebraic theory of semigroups. Vol. I*, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR**0132791** - László Fuchs,
*Infinite abelian groups. Vol. I*, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR**0255673** - Robert Gilmer,
*Commutative semigroup rings*, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1984. MR**741678** - C. Năstăsescu and F. van Oystaeyen,
*Graded ring theory*, North-Holland Mathematical Library, vol. 28, North-Holland Publishing Co., Amsterdam-New York, 1982. MR**676974** - P. Wauters,
*Factorial domains and graded rings*, Comm. Algebra**17**(1989), no. 4, 827–836. MR**990980**, DOI 10.1080/00927878908823761

## Bibliographic Information

- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**110**(1990), 21-26 - MSC: Primary 13F05; Secondary 13G05, 16A03
- DOI: https://doi.org/10.1090/S0002-9939-1990-1027092-7
- MathSciNet review: 1027092