Proving Commutativity in a Ring with s2 = s

[AKG]

Let S be a ring such that for all s in S, s² = s. Prove that S is commutative.

I've proved that for all s and t in S, (s + t)² = s² + t², and also that s + s = 0. How would I go about proving that for all s and t, st = ts? Thanks. By the way, this isn't exactly homework, I was just practicing for the GREs.

 

[Hurkyl]

What (else) is (s + t)²?

 

[AKG]

s + t = (s + t)² [by hypothesis]
s + t = s² + st + ts + t² [expanding]
s + t = s + st + ts + t [by hypothesis]
0 = st + ts [cancelling]
-st = ts [cancelling]
st = ts [since st + st = 0]

Thanks.