1. ab = 0 => ba = 0 via ba = (ba)^3 = b ab ab a = 0
> If R is a ring with unity and x^3 = x for all elements
2. cc = c => c central [i.e. xc = cx for all x]
Proof: c(x-cx)=0 so (x-cx)c=0 by 1, so xc = cxc
3. xx central via c = xx in 2.
4. cc = 2c => c central: c = ccc = 2cc central by 3.
5. x + xx central via c = x + xx in 4.
6. x = (x + xx) - xx central via 3, 5. QED
This is a special case of a famous theorem of Jacobson
The above is from my post in the thread below
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