Usual arithmetic
Modular arithmetic.
Polynomial rings.
Residue class rings
The Gaussian integers.
The kernel of the morphism Z -> Z/mZ
is mZ.
If m divides n, then there is a morphism
Z/nZ -> Z/mZ,
x + (n) -> x + (m).
Its kernel is
(m).
Fix x 
Q.
The kernel of the morphism
Q[X] -> Q,
f(X) -> f(x)
Q.
The kernel of the morphism
is (X - x).
Just like the modular arithmetic case:
If f, g are polynomials in
Q[X] such that g divides f, then there is a morphism
Q[X]/(f) -> Q[X]/(g)
x + (f) -> x + (g).
Its kernel is
(g).
The map
Z + Zi -> Z/2Z,
a + bi -> a + b
is a morphism. Check:
- 1 = 1 + 0i -> 1;
- (a + bi)(c + di) = (ac - bd) + (ad + bc)i -> ac + bd + ad + bc = (a + b)(c + d).
Its kernel is (1 + i).