http://www.merriam-webster.com/dictionary/idempotent brings the definition of idempotent, which is basically that that returns itself after the operation with itself. 

http://mathworld.wolfram.com/Ring.html brings the definition of ring, which, as you know, comes from Algebra. A ring accepts multiplication and its commutative property, as you see, so that you can swap the order of whatever you have there in your multiplication. As you do that, e comes together with e returning e and the subtraction then gives you a zero. Whatever that is inside of brackets will give you zero, and you now have only 1 left. I suppose that is the explanation.