A compactness property for prime ideals in Noetherian rings
Authors:
Clive M. Reis and T. M. Viswanathan
Journal:
Proc. Amer. Math. Soc. 25 (1970), 353-356
MSC:
Primary 13.50
DOI:
https://doi.org/10.1090/S0002-9939-1970-0254031-6
MathSciNet review:
0254031
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Abstract: A ring $R$ is compactly packed by prime ideals if whenever an ideal $I$ of $R$ is contained in the union of a family of prime ideals of $R,I$ is actually contained in one of the prime ideals of the family. It is shown that a commutative Noetherian ring is compactly packed if and only if every prime ideal is the radical of a principal ideal. For Dedekind domains this is equivalent to the torsion of the ideal class group and again to the existence of distinguished elements for the essential valuations. If a Noetherian ring $R$ is compactly packed then Krull dim. $R \leqq 1$. Also a Krull domain $R$ is compactly packed if and only if it is a Dedekind domain with torsion ideal class group.
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Keywords:
Noetherian ring,
Dedekind domain,
Krull domain,
Krull dimension
Article copyright:
© Copyright 1970
American Mathematical Society