Maps between spherical group rings

Abstract We prove that for finitely generated abelian groups $A$ A and $B$ B , the space of $\mathbb{E}_{\infty }$ E ∞ -ring maps between the spherical groups rings $\mathbb{S}[A] \to \mathbb{S}[B]$ S [ A ] → S [ B ] is equivalent to the discrete set of group homomorphisms $A \to B$ A → B . We also prove generalizations where the sphere is replaced by other ring spectra, e.g. we give a formula for the strict units in group rings of the form $R[A]$ R [ A ] for $A$ A a finite $p$ p -group and $R$ R $p$ p -completely chromatically complete.