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Proceedings of the American Mathematical Society

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



A note on semilocal rings

Author: Johnny A. Johnson
Journal: Proc. Amer. Math. Soc. 55 (1976), 469-470
MathSciNet review: 0396526
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Abstract: If $ (R,{m_1}, \ldots ,{m_w})$ is a semilocal ring whose ideal lattice is topologically complete, it is shown that: given any natural number $ n$ and any decreasing sequence $ \langle {a_i}\rangle $ of ideals of $ R$, there exists a natural number $ s\left( n \right)$ such that $ {a_{s(n)}} \subseteq (\bigcap {_i{a_i}) + {m^n}} $ where $ m = \bigcap\nolimits_{i = 1}^w {{m_i}} $. This generalizes a well-known theorem on complete semilocal rings.

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Keywords: Ideal, semilocal, rings, complete ring, Noether lattice, lattice
Article copyright: © Copyright 1976 American Mathematical Society

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