[Note: when I pressed "preview post," I got the error "LaTeX ERROR: Unknown error" all over, and most of the text didn't appear. However, it seems to be a problem with my own computer because the code is correct. Please let me know if I did something wrong, and how to fix this display problem. Thanks]

This is a seemingly-innocuous problem from my abstract algebra book that turns out to be getting the better of me. Let be a ring such that for all . Then is a commutative ring.

I've already proven that in a ring where , we have commutativity. This was done by first proving that , and then I had:

whence

But by the lemma discussed above, so we had , and so .

I figured I would do something similar for the present problem. In fact, I have already shown that . Proceeding as before, I then evaluated , but this didn't yield anything too helpful. What did work out nicely was to evaluate which led quickly to a nice result:

In fact, it seems like I'm almost done, because if I can only show that

then I can prove commutativity from there. It sounds right, and it is right, but regrettably I don't see the proof of the above fact. In spite of the progress I seem to have made, I'm stuck. Any ideas?