On independence of generators of the tautological rings

We prove that all monomials of $\kappa$-classes and $\psi$-classes are independent in $R^k(\ocM_{g,n})/R^k(\partial\ocM_{g,n})$ for all $k \leq [g/3]$. We also give a simple argument for $\kappa_l \neq 0$ in $R^l(\mathcal{M}_g)$ for $l \leq g-2$.