Ring without zero divisors

I have following question.

Is it true, that if we have a ring R (with unit) without zero divisors
satisfying that
(*) for all x there is y such that yx^3y = x,
then R is divison ring?

I can write these observations
- Two sided ideals of R are idempotent (I^2=I)
- The elements y and x^4 (as above) commute, because x^4y = xx^3y =
yx^3yx^3y = yx^3 x = yx^4.

Thank you for your ideas how to prove this or for your counterexamples!


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