x^2=1 <==> x^2-1=0 <==> (x-1)(x+1)=0 <==> x=1 or x=-1
I know this should be really easy, but I can't seem to figure it out. I don't really want an out an out answer if possible, just a starting point to work from.
The question reads as follows:
"Let R be an integral domain, and suppose satisfies . Prove that we must either have x = 1 or x = -1."
I started off by saying that if we have:
Then x has an inverse, where .
This means that x is also in the group of units of R, but then I just don't know where to go... part of me wants to ignore the commutative law and say something like:
So by right multiplication then x = 1 and . But that just seems off.
Thanks for any help.