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Proceedings of the American Mathematical Society

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

 

 

On the complete integral closure of a domain


Author: Paul Hill
Journal: Proc. Amer. Math. Soc. 36 (1972), 26-30
MSC: Primary 13G05
MathSciNet review: 0308110
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Abstract: For a given positive integer n, a semivaluation domain $ {D_n}$ is constructed so that the complete integral closure has to be applied successively exactly n times before obtaining a completely integrally closed domain. Letting $ {G_n}$ be the group of divisibility of $ {D_n}$, we set $ G = \Sigma \boxplus {G_n}$, the cardinal sum of the groups $ {G_n}$. It is concluded that the semivaluation domain D having G as its group of divisibility is a Bezout domain with the property that $ D \subset {D^ \ast } \subset {D^{ \ast \ast }} \subset {D^{ \ast \ast \ast }} \subset \cdots $ is a strictly ascending infinite chain, where $ {D^ \ast }$ is the complete integral closure of D.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0308110-7
Keywords: Complete integral closure, semivaluation, Bezout domain, group of divisibility, lattice ordered group
Article copyright: © Copyright 1972 American Mathematical Society