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)



Weakly factorial domains and groups of divisibility

Authors: D. D. Anderson and Muhammad Zafrullah
Journal: Proc. Amer. Math. Soc. 109 (1990), 907-913
MSC: Primary 13F15; Secondary 06F20, 13A05, 13A17, 13G05
MathSciNet review: 1021893
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An integral domain $ R$ is said to be weakly factorial if every nonunit of $ R$ is a product of primary elements. We give several conditions equivalent to $ R$ being weakly factorial. For example, we show that the following conditions are equivalent: (1) $ R$ is weakly factorial; (2) every convex directed subgroup of the group of divisibility of $ R$ is a cardinal summand; (3) if $ P$ is a prime ideal of $ R$ minimal over a proper principal ideal ( $ \left( x \right)$), then $ P$ has height one and $ {\left( x \right)_P} \cap R$ is principal; (4) $ R = \cap {R_P}$, where the intersection runs over the height-one primes of $ R$, is locally finite, and the $ t$-class group of $ R$ is trivial.

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

  • [1] D. D. Anderson, Star-operations induced by overrings, Comm. Alg. 16 (1988), 2535-2553. MR 955324 (89f:13032)
  • [2] D. D. Anderson and L. A. Mahaney, On primary factorizations, J. Pure Appl. Alg. 54 (1988), 141-154. MR 963540 (89h:13004)
  • [3] A. Bigard, K. Keimal, and S. Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Math., vol. 608, Springer-Verlag, New York, 1977.
  • [4] A. Bouvier, Le groupe des classes d'un anneau integré, 107$ ^{f}$ Congrés national des Sociétés savantes, Brest, 1982, sciences, fasc, IV, 85-92.
  • [5] A. Bouvier and M. Zafrullah, On some class groups of an integral domain, Bull. Soc. Math. Grèce (to appear). MR 1039430 (91g:13015)
  • [6] R. Gilmer, Multiplicative ideal theory, Marcel Dekker, New York, 1972. MR 0427289 (55:323)
  • [7] M. Griffin, Some results on $ v $-multiplication rings, Canad. J. Math. 19 (1967), 710-722. MR 0215830 (35:6665)
  • [8] I. Kaplansky, Commutative rings, revised ed., The University of Chicago Press, Chicago and London, 1974. MR 0345945 (49:10674)
  • [9] J. L. Mott, The group of divisibility and its application, Lecture Notes in Math., vol. 311, Springer-Verlag, New York, 1972, 194-208. MR 0337943 (49:2712)
  • [10] -, Convex directed subgroups of a group of divisibility, Canad. J. Math. 26 (1974), 532-542. MR 0364213 (51:468)
  • [11] J. L. Mott and M. Schexnayder, Exact sequences of semi-value groups, J. Reine Angew. Math. 283/284 (1976), 388-401. MR 0404247 (53:8050)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13F15, 06F20, 13A05, 13A17, 13G05

Retrieve articles in all journals with MSC: 13F15, 06F20, 13A05, 13A17, 13G05

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society