Proving the Commutativity of a Ring with R satisfying a^2 = a

[playa007]

Homework Statement

Let R be a ring that satisfies a^2 = a for all a in R. Prove that R is a commutative ring

Homework Equations

The Attempt at a Solution

My attempt at this solution is (ab-ba)^2 = (ba-ab)^2 is true for any ring R => (ab-ba) = (ba - ab) => 2ab = 2ba => ab = ba. The problem here is I have no method to prove that ab-ba is indeed an element of R; I'm needing help with that or a totally alternate approach to this problem is welcomed so I can perhaps gain insight

 

[Dick]

You don't need to prove ab-ba is an element of R. It's a ring. It's closed under multiplication and addition. But your method is flawed from the start. Stating that (ab-ba)^2=(ba-ab)^2 uses the property that (-1)^2=1. But (assuming the ring has a unit 1) your assumption that a^2=a for a in R would mean (-1)^2=(-1). It's a big danger in working with rings to apply algebra rules that apply to reals, but not to a general ring. Why don't you start by seeing what conclusions you can draw from (a+b)^2=a+b?