Factorization of an integrally closed ideal in two-dimensional regular local rings

Let $(R,m,k)$ be a two-dimensional regular local ring with algebraically closed residue field $k$ and $I$ be an $m$-primary integrally closed ideal in $R$. Let $T(I)$ be the set of Rees valuations of $I$ and $k(v)$ be the residue field of the valuation ring $V$ associated with $v\in T(I)$. Assume that $(a,b)$ is any minimal reduction of $I$. We show that if $I$ is the product of the distinct simple $m$-primary integrally closed ideals in $(R,m,k)$, then $k(v)$ is generated by the image of $a/b$ over $k$ for all $v\in T(I)$, and the converse of this is also true.