  Generalizing Keisler's notion of regularity for ultrafilters, Taylor
introduced degrees of regularity for ideals and showed that a countably
complete nonregular ideal on $\omega_1$ must be somewhere $\omega_1$-dense. We
prove a dichotomy about degrees of regularity for $\kappa$-complete ideals on
successor cardinals $\kappa$ and apply this to show that Taylor's Theorem does
not generalize to higher cardinals. In particular, the existence of a
nonregular ideal on $\omega_2$ does not imply the existence of an
$\omega_2$-dense ideal on $\omega_2$. We obtain similar results for normal
ideals on $\mathcal P_\kappa(\lambda)$.
