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

Quotient rings of $H\mathbb{F}_2 \wedge H\mathbb{F}_2$