Filtrations, asymptotic and Prüferian closures, cancellation laws
Authors:
Henri Dichi and Daouda Sangare
Journal:
Proc. Amer. Math. Soc. 113 (1991), 617-624
MSC:
Primary 13B22; Secondary 13A15
DOI:
https://doi.org/10.1090/S0002-9939-1991-1064901-0
MathSciNet review:
1064901
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Abstract | References | Similar Articles | Additional Information
Abstract: Let be a commutative ring. For any filtration
on the ring
let
(resp.
be the asymptotic (resp. prüferian, integral) closure of the filtration
. Then we have (*)
In this paper several examples to show that each relation in (*) may be an equality or a strict inequality even in noetherian rings, are given. Some transfer properties (such as the property that a filtration be AP or strongly AP) between the filtrations
, and
are also given and negative answers are illustrated by some examples. This paper is closed by studying some cancellation laws concerning the prüferian closure of filtrations. In particular it is shown in the main theorem that if
are filtrations on the noetherian ring
such that
, if
and if
is strongly AP then we have
. In this theorem the hypothesis "
strongly AP" cannot be weakened to "
AP" as shown in Example 2.3(3).
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1991-1064901-0
Article copyright:
© Copyright 1991
American Mathematical Society