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: 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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DOI:
https://doi.org/10.1090/S0002-9939-1991-1064901-0

Article copyright:
© Copyright 1991
American Mathematical Society