Let R be a ring and a and be exist in R where a is not equal

Let R be a ring, and a and be exist in R, where a is not equal to 0. If R is not an integral domain, give 3 examples showing that ax=b can have 0, 1 or more than 1 solution in R.

Solution

Let R =Z₆

1) a=3,b=1, ax=1 has 0 solutions (as 3 is not a unit )

2) a=5, b=1 5x=1 has 1 solution (x=5)

3) a=3, b=0 , 3x=0 has 2 solutions (x=2,x=4)