Algebraic conumdrum

[Ghoulish Delight]

I've managed to stump myself with a seemingly straight forward problem. Find all valid solutions for a and b in the following expression:

a*b=a

Now, at first the solution is obvious. b=a/a, b=1. So the set of soultions is:

a={all numbers}
b={1}

So far so good. However, a quick, non-algebraic analysis of the expression clearly shows there is another solution set, namely:

a={0}
b={all numbers}

So why can't I get the expression to yield that solution algebraicly? It's obvious looking at it, but I can't find a systematic way of solving the equation to get a=0. What am I missing (and am I going to feel really stupid when someone points it out to me)?