Morphisms relate rings modulo an ideal. Let R and S be commutative rings. The image of a morphism R -> S can be entirely described in terms of R.
First isomorphism theorem
If f : R -> S is a morphism of rings, then R/Ker(f) is isomorphic to the image f(R).
So the image is a residue class ring of R.
Conversely, if I is and ideal of R, then the morphism
has kernel I. This means that every ideal occurs as the kernel of a morphism.
