  Strata of $k$-differentials on smooth curves parameterize sections of the
$k$-th power of the canonical bundle with prescribed orders of zeros and poles.
Define the tautological ring of the projectivized strata using the $\kappa$ and
$\psi$ classes of moduli spaces of pointed smooth curves along with the
tautological class $\eta$ of the Hodge bundle. We show that if there is no pole
of order $k$, then the tautological ring is generated by $\eta$ only, and
otherwise it is generated by the $\psi$ classes corresponding to the poles of
order $k$.
