Hosszú’s equation over the Gaussian and Eisenstein integers

Summary.

We consider Hosszú’s famous functional equation f(x) + f(y) = f(xy) + f(x + y − xy). We completely describe the set of functions $f : R \rightarrow {\mathbb{A}}$ satisfying this equation, where R is the set of the Gaussian or Eisenstein integers and ${\mathbb{A}}$ is an arbitrary abelian group.