A theorem of Briançon-Skoda type for regular local rings containing a field

Let $(R,m)$ be a regular local ring containing a field. We give a refinement of the Briançon-Skoda theorem showing that if $J$ is a minimal reduction of $I$ where $I$ is $m$-primary, then $\overline{I^{d+w}} \subseteq J^{w+1}\mathfrak{a}$ where $d = \dim R$ and $\mathfrak{a}$ is the largest ideal such that $\mathfrak{a} J = \mathfrak{a} I$. The proof uses tight closure in characteristic $p$ and reduction to characteristic $p$ for rings containing the rationals.