  In this paper, we prove the multiplicativity of the K\"unneth spectral
sequence. This is established by an analogue of the Comparison Theorem from
homological algebra, which we suspect may be useful for other spectral
sequences. This multiplicativity is then used to compute the action of the
Dyer-Lashof algebra on $\HF_p \wedge_{ku} \HF_p$, $\HF_p \wedge_{\BP} \HF_p$,
and part of the action on $\HF_p \wedge_{MU} \HF_p$. We then relate these
computations to the construction of commutative $R$-algebra structures on
commutative $\HF_p$-algebras. In the case of $MU$, we obtain a necessary
closure condition on ideals $I\subset MU_*$ such that $MU/I$ can be realized as
a commutative $MU$-algebra.
