Reductions of ideals in local rings with finite residue fields

Let $I$ be a proper nonnilpotent ideal in a local (Noetherian) ring $(R,M)$ and let $J$ be a reduction of $I$; that is, $J$ $\subseteq$ $I$ and $JI^n$ $=$ $I^{n+1}$ for some nonnegative integer $n$. We prove that there exists a finite free local unramified extension ring $S$ of $R$ such that the ideal $IS$ has a minimal reduction $K$ $\subseteq$ $JS$ with the property that the number of elements in a minimal generating set of $K$ is equal to the analytic spread of $K$ and thus also equal to the analytic spread of $I$.