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Proceedings of the American Mathematical Society
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
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1021893-7
PII: S 0002-9939(1990)1021893-7
Article copyright: © Copyright 1990 American Mathematical Society