In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.

More explicitly, if R and S are rings, then a ring homomorphism is a function f : R → S such that

(Additive inverses and the additive identity are part of the structure too, but it is not necessary to require explicitly that they too are respected, because these conditions are consequences of the three conditions above. On the other hand, neglecting to include the condition f(1_R) = 1_S would cause several of the properties below to fail.)