Let $A$ be a perfect ring of characteristic $p>0$. If $x\in A$ is a nonzero divisor then $$ (x^{1/p^{\infty}})=\bigcup_{e=1}^{\infty}(x^{1/p^{e}})A $$ is an $A$-flat ideal.
Reading Hochster's Canonical Elements in Local Cohomology Modules and the Direct Summand Conjecture, I came across with this statement above and and I have been stuck for days trying to prove it.
Any help would be appreciated.