  Let $R$ denote a commutative Noetherian ring, $I$ an ideal of $R$, and let
$S$ be a multiplicatively closed subset of $R$. In \cite{Ra1}, Ratliff showed
that the sequence of sets ${\rm Ass}_RR/\bar{I}\subseteq {\rm Ass}_RR/\bar{I^2}
\subseteq {\rm Ass}_R R/\bar{I^3}\subseteq \dots $ increases and eventually
stabilizes to a set denoted $\bar{A^\ast}(I)$. In \cite{Mc2}, S. McAdam gave an
interesting description of $\bar{A^\ast}(I)$ by making use of $R[It,t^{-1}]$,
the Rees ring of $I$. In this paper, we give a second description of
$\bar{A^\ast}(I)$ by making use of the Rees valuation rings of $I$. We also
reprove a result concerning when $\bar{I^n}R_S\cap R=\bar{I^n}$ for all
integers $n>0$.
