Maps between spherical group rings

We prove that for finitely generated abelian groups A and B, the space of E_∞-ring maps between the spherical groups rings S[A] → S[B] is equivalent to the discrete set of group homomorphisms 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 $R[A]$ for A a finite p-group and R p-completely chromatically complete.