Let $R$ be a Noetherian ring, and let $I$ be an ideal of $R$. Fix a rational number $a=\frac{p}{q}$ with $p, q\in \mathbb{Z_\geq 0}$ $q\neq 0$. We deﬁne $I_a = \{x \in R: x^q\in \overline{I^p}\}$, where overline denotes integral closure.

Huneke-Swanson show, (in their book on Integral Closure), that $I_a$ is well defined, is indeed an ideal and is integrally closed. If $a\leq b$, $I_b\subseteq I_a$ and if $n$ is a positive integer, then, $I_n=\overline{I^n}$.

I was curious to know what else is known about these rational powers. Are there any computations done in special cases (e.g. monomial ideals)? Do these ideals find any use other than computing the integral closure of the extended Rees ring in the Laurent polynomial ring? I would appreciate if anyone has any references on this subject other than Huneke-Swanson book.

So far I have only found a paper by David Rush "Rees valuations and asymptotic primes of rational powers in Noetherian rings and lattices"

~~PS: I am not sure why, but I am unable to typeset curly braces (I am using backslash followed by brace sign, but it does not show up) and the "less than" (causes text after the symbol to vanish) symbol.~~