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Nonstandard theory of Zariski rings


Author: Loren C. Larson
Journal: Proc. Amer. Math. Soc. 29 (1971), 23-29
MSC: Primary 13.25; Secondary 02.00
DOI: https://doi.org/10.1090/S0002-9939-1971-0279082-8
MathSciNet review: 0279082
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Abstract: Let $ ^ \ast R$ be an enlargement (in the sense of A. Robinson) of a Zariski ring (R, A), let $ \mu $ be the monad of zero in $ ^ \ast R$ when R is given the A-adic topology and set $ {R_\mu }$ equal to the quotient ring $ ^ \ast R/\mu $. It is shown that $ (R,{R_\mu })$ is a flat couple, and $ {R_\mu }$ is Noetherian if and only if it is semilocal. Furthermore, if R is semilocal and A is the (Jacobson) radical then $ {R_\mu }$ is semilocal, with the same number of maximal ideals and the same (Krull) dimension as R.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0279082-8
Keywords: Zariski rings, semilocal rings, A-adic rings, ring completions, ring extensions, flat couples of rings, nonstandard models, enlargements, ultraproducts
Article copyright: © Copyright 1971 American Mathematical Society

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