Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



Dedekind domains and graded rings

Authors: Fabien Decruyenaere and Eric Jespers
Journal: Proc. Amer. Math. Soc. 110 (1990), 21-26
MSC: Primary 13F05; Secondary 13G05, 16A03
MathSciNet review: 1027092
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

  • [1] D. D. Anderson and D. F. Anderson, Divisibility properties of graded domains, Canad. J. Math. 24 (1982), 196-215. MR 650859 (84e:13026)
  • [2] D. F. Anderson, Graded Krull domains, Comm. Algebra 7 (1979), 79-106. MR 514866 (80c:13015)
  • [3] A. J. Clifford and G. B. Preston, The algebraic theory of semigroups, vol. I, Math. Surv. Mono. vol. 7, Amer. Math. Soc., Providence, RI, 1961. MR 0132791 (24:A2627)
  • [4] L. Fuchs, Infinite abelian groups, vol. II, Academic Press, New York, 1970. MR 0255673 (41:333)
  • [5] R. Gilmer, Commutative semigroup rings, The University of Chicago Press, Chicago and London, 1984. MR 741678 (85e:20058)
  • [6] C. Nastasescu and F. Van Oystaeyen, Graded ring theory, North-Holland, Amsterdam, 1982. MR 676974 (84i:16002)
  • [7] P. Wauters, Factorial domains and graded rings, Comm. Algebra (to appear). MR 990980 (90d:13017)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13F05, 13G05, 16A03

Retrieve articles in all journals with MSC: 13F05, 13G05, 16A03

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society