n. a mapping from one algebraic structure to another under which the structural properties of its domain are preserved in its range in the sense that if * is the operation on the domain, and ° is the operation on the range, then
θ(x * y) = θ(x) ° θ(y).
In particular, a group homomorphism is a mapping θ such that both domain and range are groups, and
θ(xy) = θ(x) θ(y)
for all x and y in the domain; a ring homomorphism is a mapping θ from one ring to another such that
θ(x + y) = θ(x) + θ(y) and θ(xy) = θ(x) θ(y)
for all x and y in the domain; a module homomorphism is a mapping such that
θ(x + y) = θ(x) + θ(y) and θ(rx) = r θ(x)
for all x and y in the R-module and r in the ring R (where, if R is a field, then θ is a linear mapping). In group theory, homomorphisms are taken to be surjective unless otherwise stated to the contrary. The natural epimorphism or natural homomorphism is the homomorphism ν from G to the factor group G/K, that is defined for rings and modules by ν(x) = x + K and for groups by ν(x) = xK. See also isomorphism, epimorphism, monomorphism, morphism, endomorphism.