Here's the formulation for group theory; replace "group" by whatever to get it in other domains (it usually holds whenever it may be formulated).
Let f:G->H be a homomorphism. Then Ker f is a normal subgroup of G, and G/Ker f is isomorphic to Im f.
In fact, since for any normal subgroup N of G the map F: G -> G/N defined by F(g) = gN is a homomorphism, this shows that kernels and normal subgroups are really the same!