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*) = *g***N** is a homomorphism, this shows that kernels and normal subgroups are really the same!