  We prove a conjecture of J.P. May concerning the nilpotence of elements in
ring spectra with power operations, i.e., $H_\infty$-ring spectra. Using an
explicit nilpotence bound on the torsion elements in $K(n)$-local
$H_\infty$-algebras over $E_n$, we reduce the conjecture to the nilpotence
theorem of Devinatz, Hopkins, and Smith. As corollaries we obtain nilpotence
results in various bordism rings including $M\mathit{Spin}_*$ and
$M\mathit{String}_*$, results about the behavior of the Adams spectral sequence
for $E_\infty$-ring spectra, and the non-existence of $E_\infty$-ring
structures on certain complex oriented ring spectra.
