LetK be a ring with an identity 1 ≠ 0 andM, L two unitaryK-modules. Then, for any additive mappingf:M →L, the setH f :={α ∈ K ∣ f(αx)=αf(x) for allx ∈ M} forms a subring ofK, the ‘homogeneity ring’ off. It is shown that, forM ≠ {0},L ≠ {0} and any subringS ofK for whichM is a freeS-module, there exists an additive mappingf:M→L such thatH f =S. This result is applied to the four Cauchy functional equations, and it leads also to an answer to the question as to whether it is possible to introduce onM a multiplication ·:M × M → M makingM into a ring but not into aK-algebra.