  We study modules over the commutative ring spectrum $H\mathbb F_2\wedge
H\mathbb F_2$, whose coefficient groups are quotients of the dual Steenrod
algebra by collections of the Milnor generators. We show that very few of these
quotients admit algebra structures, but those that do can be constructed
simply: killing a generator $\xi_k$ in the category of associative algebras
freely kills the higher generators $\xi_{k+n}$. Using new information about the
conjugation operation in the dual Steenrod algebra, we also consider quotients
by families of Milnor generators and their conjugates. This allows us to
produce a family of associative $H\mathbb F_2\wedge H\mathbb F_2$-algebras
whose coefficient rings are finite-dimensional and exhibit unexpected duality
features. We then use these algebras to give detailed computations of the
homotopy groups of several modules over this ring spectrum.
