The ring of quaternions, discovered by W.R. Hamilton provides the most often cited example of a skew field or division ring it consists of all possible sums of the form
a1 + bi + cj +dk
where a,b,c and d are real numbers and 1,i,j and k are members of the quaternionic group. For concreteness we can take 1,i,j and k to be 4 x 4 matrices:
1 = 1 0 0 0 i = 0 -1 0 0 j = 0 0 -1 0 k = 0 0 0 -1
0 1 0 0 1 0 0 0 0 0 0 1 0 0 -1 0
0 0 1 0 0 0 0 -1 1 0 0 0 0 1 0 0
0 0 0 1 0 0 1 0 0 -1 0 0 1 0 0 0