QUOTIENT RINGS OF HF₂ ∧ HF₂

We study modules over the commutative ring spectrum HF₂ ∧ HF₂, 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 ξ_k in the category of associative algebras freely kills the higher generators ξ_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 HF₂ ∧ HF₂-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.