Thanks. It look so easy when you see it

I'm now stuck with

3) This time suppose

for all

in a ring with identity

Use 1) to prove that

is commutative.

__Hint__:

__Spoiler__:

It's immediate that

is an idempotent, and thus central by 1), for all

Put

to show that

is central for all

As suggested in the hint - I could prove the following:

1.

2.

3.

How do I show R is commutative? A push again plz?

Infact I was trying to show that ab+ba=0. Then using (1) above I will show

ab = ba and hence will be done. But no success.