Filtrations, asymptotic and Prüferian closures, cancellation laws

Let $A$ be a commutative ring. For any filtration $f = ({I_n})$ on the ring $A$ let $\bar f$ (resp. $P(f),f')$ be the asymptotic (resp. prüferian, integral) closure of the filtration $f$. Then we have (*) $f \leq f' \leq P(f) \leq \bar f.$ 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 $f,P(f)$, and $\bar f$ 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 $f,g,h$ are filtrations on the noetherian ring $A$ such that $\sqrt f \subseteq \sqrt h$, if $P(fh) \leq P(gh)$ and if $h$ is strongly AP then we have $P(f) \leq P(g)$. In this theorem the hypothesis "$h$ strongly AP" cannot be weakened to "$h$ AP" as shown in Example 2.3(3).